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Theorem pw1eq 4143
 Description: Equality theorem for unit power class. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
pw1eq (A = B1A = 1B)

Proof of Theorem pw1eq
StepHypRef Expression
1 pweq 3725 . . 3 (A = BA = B)
21ineq1d 3456 . 2 (A = B → (A ∩ 1c) = (B ∩ 1c))
3 df-pw1 4137 . 2 1A = (A ∩ 1c)
4 df-pw1 4137 . 2 1B = (B ∩ 1c)
52, 3, 43eqtr4g 2410 1 (A = B1A = 1B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3208  ℘cpw 3722  1cc1c 4134  ℘1cpw1 4135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-pw1 4137 This theorem is referenced by:  pw10b  4166  pw1equn  4331  pw1eqadj  4332  sspw1  4335  sspw12  4336  addceq2  4384  ncfinraise  4481  ncfinlower  4483  nnpw1ex  4484  tfindi  4496  tfinsuc  4498  sfin01  4528  sfindbl  4530  1cvsfin  4542  vfinspsslem1  4550  rnsi  5521  pw1fnval  5851  pw1fnf1o  5855  enpw1  6062  ncpw1c  6154  ncspw1eu  6159  pw1eltc  6162  tcdi  6164  ce0nnul  6177  ce0nnuli  6178  ce0addcnnul  6179  cenc  6181  ce0nulnc  6184  ce2  6192  elcan  6329
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