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Theorem snexALT 5389
Description: Alternate proof of snex 5442 using Power Set (ax-pow 5371) instead of Pairing (ax-pr 5438). Unlike in the proof of zfpair 5427, Replacement (ax-rep 5285) 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 4849 . . 3 {𝐴} ⊆ 𝒫 𝐴
2 ssexg 5329 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
31, 2mpan 690 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4 pwexg 5384 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
54con3i 154 . . 3 (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6 snprc 4722 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 216 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
8 0ex 5313 . . . 4 ∅ ∈ V
97, 8eqeltrdi 2847 . . 3 𝐴 ∈ V → {𝐴} ∈ V)
105, 9syl 17 . 2 (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
113, 10pm2.61i 182 1 {𝐴} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632
This theorem is referenced by:  p0exALT  5391
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