Theorem snexALT 5275

Description: Alternate proof of snex 5323 using Power Set (ax-pow 5258) instead of Pairing (ax-pr 5321). Unlike in the proof of zfpair 5313, Replacement (ax-rep 5182) 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 4768	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5219	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 688	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5271	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 157	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4646	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 218	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5203	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2921	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
10	5, 9	syl 17	. 2 ⊢ (¬ 𝒫 𝐴 ∈ V → {𝐴} ∈ V)
11	3, 10	pm2.61i 184	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561

This theorem is referenced by:  p0exALT  5277