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Theorem ssunieq 3604
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 3599 . . 3 (A BA B)
2 unissb 3601 . . . 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 2954 . 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 1242   wcel 1390  wral 2300  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-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-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  unimax  3605
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