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Theorem unidif 3582
 Description: If the difference A ∖ B 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 (x A y (AB)xy (AB) = A)
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 3581 . . 3 (x A y (AB)xy A (AB))
2 difss 3043 . . . 4 (AB) ⊆ A
32unissi 3573 . . 3 (AB) ⊆ A
41, 3jctil 295 . 2 (x A y (AB)xy → ( (AB) ⊆ A A (AB)))
5 eqss 2933 . 2 ( (AB) = A ↔ ( (AB) ⊆ A A (AB)))
64, 5sylibr 137 1 (x A y (AB)xy (AB) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226  ∀wral 2280  ∃wrex 2281   ∖ cdif 2887   ⊆ wss 2890  ∪ cuni 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-uni 3551 This theorem is referenced by: (None)
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