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

Step	Hyp	Ref	Expression
1		snsspw 4843	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5322	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 690	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5377	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 154	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4716	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 216	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5306	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2848	. . 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 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