Theorem snexALT 5353

Description: Alternate proof of snex 5409 using Power Set (ax-pow 5335) instead of Pairing (ax-pr 5403). Unlike in the proof of zfpair 5391, Replacement (ax-rep 5237) 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 4808	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5289	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 702	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5348	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 155	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4682	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5269	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2870	. . 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

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4563  df-sn 4589

This theorem is used by:  p0exALT  5355