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Theorem ssunieq 3769
Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)
Assertion
Ref Expression
ssunieq ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssunieq
StepHypRef Expression
1 elssuni 3764 . . 3 (𝐴𝐵𝐴 𝐵)
2 unissb 3766 . . . 4 ( 𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
32biimpri 132 . . 3 (∀𝑥𝐵 𝑥𝐴 𝐵𝐴)
41, 3anim12i 336 . 2 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → (𝐴 𝐵 𝐵𝐴))
5 eqss 3112 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
64, 5sylibr 133 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wral 2416  wss 3071   cuni 3736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737
This theorem is referenced by:  unimax  3770  hashinfuni  10535  hashennnuni  10537
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