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Mirrors > Home > MPE Home > Th. List > elALT | Structured version Visualization version GIF version |
Description: Alternate proof of el 5261, shorter but requiring more axioms. (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 3496 | . . 3 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4593 | . 2 ⊢ 𝑥 ∈ {𝑥} |
3 | snex 5322 | . . 3 ⊢ {𝑥} ∈ V | |
4 | eleq2 2899 | . . 3 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥})) | |
5 | 3, 4 | spcev 3605 | . 2 ⊢ (𝑥 ∈ {𝑥} → ∃𝑦 𝑥 ∈ 𝑦) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1774 ∈ wcel 2108 {csn 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-dif 3937 df-un 3939 df-nul 4290 df-sn 4560 df-pr 4562 |
This theorem is referenced by: (None) |
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