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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 2763 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2758 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylib 122 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-v 2754 |
This theorem is referenced by: elex22 2767 elex2 2768 ceqsalt 2778 ceqsalg 2780 cgsexg 2787 cgsex2g 2788 cgsex4g 2789 vtoclgft 2802 vtocleg 2823 vtoclegft 2824 spc2egv 2842 spc2gv 2843 spc3egv 2844 spc3gv 2845 eqvincg 2876 tpid3g 3722 iinexgm 4172 copsex2t 4263 copsex2g 4264 ralxfr2d 4482 rexxfr2d 4483 fliftf 5821 eloprabga 5983 ovmpt4g 6019 spc2ed 6258 eroveu 6652 supelti 7031 genpassl 7553 genpassu 7554 eqord1 8470 nn1suc 8968 bj-inex 15117 |
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