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| 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.) | 
| Ref | Expression | 
|---|---|
| snexALT | ⊢ {𝐴} ∈ V | 
| 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 | 
| Colors of variables: wff setvar class | 
| 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 | 
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