Theorem snexALT 5344

Description: Alternate proof of snex 5396 using Power Set (ax-pow 5326) instead of Pairing (ax-pr 5390). Unlike in the proof of zfpair 5382, Replacement (ax-rep 5231) 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 4803	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
2		ssexg 5280	. . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
3	1, 2	mpan 703	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
4		pwexg 5339	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
5	4	con3i 155	. . 3 ⊢ (¬ 𝒫 𝐴 ∈ V → ¬ 𝐴 ∈ V)
6		snprc 4677	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8		0ex 5260	. . . 4 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2868	. . 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 1570   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4279  df-pw 4558  df-sn 4584

This theorem is used by:  p0exALT  5346