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Theorem ssequn2 3217
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 3214 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 uncom 3188 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2123 . 2 ((𝐴𝐵) = 𝐵 ↔ (𝐵𝐴) = 𝐵)
41, 3bitri 183 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  cun 3037  wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by:  unabs  3275  pwssunim  4174  pwundifss  4175  oneluni  4321  relresfld  5036  relcoi1  5038  fsnunf  5586  unsnfidcel  6775  fidcenumlemr  6809  exmidfodomrlemim  7021  ennnfonelemhf1o  11832
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