Theorem snexALT 5306

Description: Alternate proof of snex 5354 using Power Set (ax-pow 5288) instead of Pairing (ax-pr 5352). Unlike in the proof of zfpair 5344, Replacement (ax-rep 5209) 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 4775	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5247	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 687	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5301	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 154	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4653	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 215	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5231	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2847	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
10	5, 9	syl 17	. 2 ⊢ (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
11	3, 10	pm2.61i 182	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562

This theorem is referenced by:  p0exALT  5308