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Theorem snexALT 4882
Description: Alternate proof of snex 4938 using Power Set (ax-pow 4873) instead of Pairing (ax-pr 4936). Unlike in the proof of zfpair 4934, Replacement (ax-rep 4804) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT {𝐴} ∈ V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 4407 . . 3 {𝐴} ⊆ 𝒫 𝐴
2 ssexg 4837 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
31, 2mpan 706 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4 pwexg 4880 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
54con3i 150 . . 3 (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6 snprc 4285 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
8 0ex 4823 . . . 4 ∅ ∈ V
97, 8syl6eqel 2738 . . 3 𝐴 ∈ V → {𝐴} ∈ V)
105, 9syl 17 . 2 (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
113, 10pm2.61i 176 1 {𝐴} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211
This theorem is referenced by:  p0exALT  4884
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