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Theorem unidif 4437
Description: If the difference 𝐴𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 4436 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 𝐴 (𝐴𝐵))
2 difss 3715 . . . 4 (𝐴𝐵) ⊆ 𝐴
32unissi 4427 . . 3 (𝐴𝐵) ⊆ 𝐴
41, 3jctil 559 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 → ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
5 eqss 3598 . 2 ( (𝐴𝐵) = 𝐴 ↔ ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
64, 5sylibr 224 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wral 2907  wrex 2908  cdif 3552  wss 3555   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-uni 4403
This theorem is referenced by:  ordunidif  5732
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