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| Mirrors > Home > MPE Home > Th. List > elALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of el 5441, shorter but requiring ax-sep 5295. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| elALT | ⊢ ∃𝑦 𝑥 ∈ 𝑦 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3483 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | selsALT 5443 | . 2 ⊢ (𝑥 ∈ V → ∃𝑦 𝑥 ∈ 𝑦) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∃wex 1778 ∈ wcel 2107 Vcvv 3479 | 
| 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-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: (None) | 
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