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Theorem ssunieq 3854
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 3849 . . 3 (𝐴𝐵𝐴 𝐵)
2 unissb 3851 . . . 4 ( 𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
32biimpri 133 . . 3 (∀𝑥𝐵 𝑥𝐴 𝐵𝐴)
41, 3anim12i 338 . 2 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → (𝐴 𝐵 𝐵𝐴))
5 eqss 3182 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
64, 5sylibr 134 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wral 2465  wss 3141   cuni 3821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822
This theorem is referenced by:  unimax  3855  hashinfuni  10770  hashennnuni  10772
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