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Theorem snexALT 5382
Description: Alternate proof of snex 5435 using Power Set (ax-pow 5364) instead of Pairing (ax-pr 5431). Unlike in the proof of zfpair 5420, Replacement (ax-rep 5278) 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 4843 . . 3 {𝐴} ⊆ 𝒫 𝐴
2 ssexg 5322 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
31, 2mpan 690 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4 pwexg 5377 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
54con3i 154 . . 3 (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6 snprc 4716 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 216 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
8 0ex 5306 . . . 4 ∅ ∈ V
97, 8eqeltrdi 2848 . . 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 1539  wcel 2107  Vcvv 3479  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-pw 4601  df-sn 4626
This theorem is referenced by:  p0exALT  5384
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