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Theorem unidif 4947
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 4946 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 𝐴 (𝐴𝐵))
2 difss 4132 . . . 4 (𝐴𝐵) ⊆ 𝐴
32unissi 4918 . . 3 (𝐴𝐵) ⊆ 𝐴
41, 3jctil 521 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 → ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
5 eqss 3998 . 2 ( (𝐴𝐵) = 𝐴 ↔ ( (𝐴𝐵) ⊆ 𝐴 𝐴 (𝐴𝐵)))
64, 5sylibr 233 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wral 3062  wrex 3071  cdif 3946  wss 3949   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-uni 4910
This theorem is referenced by:  ordunidif  6414
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