Theorem snexALT 5358

Description: Alternate proof of snex 5414 using Power Set (ax-pow 5340) instead of Pairing (ax-pr 5408). Unlike in the proof of zfpair 5396, Replacement (ax-rep 5243) 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 4814	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5297	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 702	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5353	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 155	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4688	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5275	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2878	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
10	5, 9	syl 18	. 2 ⊢ (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
11	3, 10	pm2.61i 184	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1568   ∈ wcel 2150  Vcvv 3462   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595

This theorem is referenced by:  p0exALT  5360