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Theorem ssequn2 3171
Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
Assertion
Ref Expression
ssequn2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)

Proof of Theorem ssequn2
StepHypRef Expression
1 ssequn1 3168 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 uncom 3142 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2095 . 2 ((𝐴𝐵) = 𝐵 ↔ (𝐵𝐴) = 𝐵)
41, 3bitri 182 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1289  cun 2995  wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010
This theorem is referenced by:  unabs  3228  pwssunim  4102  pwundifss  4103  oneluni  4249  relresfld  4947  relcoi1  4949  fsnunf  5480  unsnfidcel  6611  fidcenumlemr  6643  exmidfodomrlemim  6806
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