Theorem pw1eq 4144

Description: Equality theorem for unit power class. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
pw1eq	⊢ (A = B → ℘1A = ℘1B)

Proof of Theorem pw1eq

Step	Hyp	Ref	Expression
1		pweq 3726	. . 3 ⊢ (A = B → ℘A = ℘B)
2	1	ineq1d 3457	. 2 ⊢ (A = B → (℘A ∩ 1c) = (℘B ∩ 1c))
3		df-pw1 4138	. 2 ⊢ ℘1A = (℘A ∩ 1c)
4		df-pw1 4138	. 2 ⊢ ℘1B = (℘B ∩ 1c)
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ℘1A = ℘1B)

Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3209  ℘cpw 3723  1_cc1c 4135  ℘₁cpw1 4136

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725  df-pw1 4138

This theorem is referenced by:  pw10b  4167  pw1equn  4332  pw1eqadj  4333  sspw1  4336  sspw12  4337  addceq2  4385  ncfinraise  4482  ncfinlower  4484  nnpw1ex  4485  tfindi  4497  tfinsuc  4499  sfin01  4529  sfindbl  4531  1cvsfin  4543  vfinspsslem1  4551  rnsi  5522  pw1fnval  5852  pw1fnf1o  5856  enpw1  6063  ncpw1c  6155  ncspw1eu  6160  pw1eltc  6163  tcdi  6165  ce0nnul  6178  ce0nnuli  6179  ce0addcnnul  6180  cenc  6182  ce0nulnc  6185  ce2  6193  elcan  6330