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Theorem eqpw1 4162
Description: A condition for equality to unit power class. (Contributed by SF, 21-Jan-2015.)
Assertion
Ref Expression
eqpw1 1 1c
Distinct variable groups:   ,   ,

Proof of Theorem eqpw1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pw1ss1c 4158 . . 3 1 1c
2 sseq1 3292 . . 3 1 1c 1 1c
31, 2mpbiri 224 . 2 1 1c
4 ssofeq 4077 . . . 4 1c 1 1c 1 1c 1
51, 4mpan2 652 . . 3 1c 1 1c 1
6 df-ral 2619 . . . . 5 1c 1 1c 1
7 el1c 4139 . . . . . . . . 9 1c
87imbi1i 315 . . . . . . . 8 1c 1 1
9 19.23v 1891 . . . . . . . 8 1 1
108, 9bitr4i 243 . . . . . . 7 1c 1 1
1110albii 1566 . . . . . 6 1c 1 1
12 alcom 1737 . . . . . 6 1 1
1311, 12bitr4i 243 . . . . 5 1c 1 1
146, 13bitri 240 . . . 4 1c 1 1
15 snex 4111 . . . . . . 7
16 eleq1 2413 . . . . . . . 8
17 eleq1 2413 . . . . . . . 8 1 1
1816, 17bibi12d 312 . . . . . . 7 1 1
1915, 18ceqsalv 2885 . . . . . 6 1 1
20 snelpw1 4146 . . . . . . 7 1
2120bibi2i 304 . . . . . 6 1
2219, 21bitri 240 . . . . 5 1
2322albii 1566 . . . 4 1
2414, 23bitri 240 . . 3 1c 1
255, 24syl6bb 252 . 2 1c 1
263, 25biadan2 623 1 1 1c
Colors of variables: wff set class
Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  wral 2614   wss 3257  csn 3737  1cc1c 4134  1 cpw1 4135
This theorem is referenced by:  eqpw1relk  4479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137
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