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Theorem ssequn2 3276
 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 3273 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 uncom 3247 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2162 . 2 ((𝐴𝐵) = 𝐵 ↔ (𝐵𝐴) = 𝐵)
41, 3bitri 183 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∪ cun 3096   ⊆ wss 3098 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111 This theorem is referenced by:  unabs  3334  pwssunim  4239  pwundifss  4240  oneluni  4386  relresfld  5108  relcoi1  5110  fsnunf  5660  unsnfidcel  6854  fidcenumlemr  6888  exmidfodomrlemim  7115  ennnfonelemhf1o  12093
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