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Theorem unidif 3603
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 (x A y (AB)xy (AB) = A)
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 3602 . . 3 (x A y (AB)xy A (AB))
2 difss 3064 . . . 4 (AB) ⊆ A
32unissi 3594 . . 3 (AB) ⊆ A
41, 3jctil 295 . 2 (x A y (AB)xy → ( (AB) ⊆ A A (AB)))
5 eqss 2954 . 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 1242  wral 2300  wrex 2301  cdif 2908  wss 2911   cuni 3571
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by: (None)
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