Firmly established or dug in so deeply that something is very hard to change or remove.
From Old French 'trenche' (cut, trench); originally meant literally digging protective ditches in military warfare, then evolved to mean any deeply established position.
The word comes directly from WWI trench warfare—soldiers literally dug trenches so deep and complex that enemies couldn't dislodge them, making 'entrenched' the perfect metaphor for ideas that won't budge.
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