Theorem elpw 3728

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
elpw.1	⊢ A ∈ V

Assertion

Ref	Expression
elpw	⊢ (A ∈ ℘B ↔ A ⊆ B)

Proof of Theorem elpw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw.1	. 2 ⊢ A ∈ V
2		sseq1 3292	. 2 ⊢ (x = A → (x ⊆ B ↔ A ⊆ B))
3		df-pw 3724	. 2 ⊢ ℘B = {x ∣ x ⊆ B}
4	1, 2, 3	elab2 2988	1 ⊢ (A ∈ ℘B ↔ A ⊆ B)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722

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

This theorem is referenced by:  elpwg  3729  prsspw  3878  pwpr  3883  pwtp  3884  pwv  3886  sspwuni  4051  iinpw  4054  iunpwss  4055  snelpwg  4114  snelpwi  4116  unipw  4117  sspwb  4118  pwadjoin  4119  elpw1  4144  dfpw2  4327  eqpw1relk  4479  nnadjoinpw  4521  tfinnnlem1  4533  pw1fnf1o  5855  mapexi  6003  mapval2  6018  mapsspm  6021  mapsspw  6022  enpw1pw  6075  enprmaplem5  6080  enprmaplem6  6081