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Mirrors > Home > ILE Home > Th. List > elisset | GIF version |
Description: An element of a class exists. (Contributed by NM, 1-May-1995.) |
Ref | Expression |
---|---|
elisset | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2692 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylib 121 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 |
This theorem is referenced by: elex22 2701 elex2 2702 ceqsalt 2712 ceqsalg 2714 cgsexg 2721 cgsex2g 2722 cgsex4g 2723 vtoclgft 2736 vtocleg 2757 vtoclegft 2758 spc2egv 2775 spc2gv 2776 spc3egv 2777 spc3gv 2778 eqvincg 2809 tpid3g 3638 iinexgm 4079 copsex2t 4167 copsex2g 4168 ralxfr2d 4385 rexxfr2d 4386 fliftf 5700 eloprabga 5858 ovmpt4g 5893 spc2ed 6130 eroveu 6520 supelti 6889 genpassl 7332 genpassu 7333 eqord1 8245 nn1suc 8739 bj-inex 13105 |
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