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Theorem ssunieq 3587
 Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)
Assertion
Ref Expression
ssunieq ((A B x B xA) → A = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq
StepHypRef Expression
1 elssuni 3582 . . 3 (A BA B)
2 unissb 3584 . . . 4 ( BAx B xA)
32biimpri 124 . . 3 (x B xA BA)
41, 3anim12i 321 . 2 ((A B x B xA) → (A B BA))
5 eqss 2937 . 2 (A = B ↔ (A B BA))
64, 5sylibr 137 1 ((A B x B xA) → A = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ∀wral 2284   ⊆ wss 2894  ∪ cuni 3554 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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555 This theorem is referenced by:  unimax  3588
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