Theorem snexALT 5342

Description: Alternate proof of snex 5398 using Power Set (ax-pow 5324) instead of Pairing (ax-pr 5392). Unlike in the proof of zfpair 5380, Replacement (ax-rep 5229) 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

Step	Hyp	Ref	Expression
1		snsspw 4804	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5281	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 700	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5337	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 154	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4678	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 218	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5259	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2872	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
10	5, 9	syl 17	. 2 ⊢ (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
11	3, 10	pm2.61i 183	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4288  df-pw 4559  df-sn 4585

This theorem is referenced by:  p0exALT  5344