Theorem snexALT 5381

Description: Alternate proof of snex 5431 using Power Set (ax-pow 5363) instead of Pairing (ax-pr 5427). Unlike in the proof of zfpair 5419, 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

Step	Hyp	Ref	Expression
1		snsspw 4845	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5323	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 687	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5376	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 154	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4721	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 215	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5307	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2840	. . 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 1540   ∈ wcel 2105  Vcvv 3473   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629

This theorem is referenced by:  p0exALT  5383