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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 2741 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2736 | . 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 1348 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 |
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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: elex22 2745 elex2 2746 ceqsalt 2756 ceqsalg 2758 cgsexg 2765 cgsex2g 2766 cgsex4g 2767 vtoclgft 2780 vtocleg 2801 vtoclegft 2802 spc2egv 2820 spc2gv 2821 spc3egv 2822 spc3gv 2823 eqvincg 2854 tpid3g 3696 iinexgm 4138 copsex2t 4228 copsex2g 4229 ralxfr2d 4447 rexxfr2d 4448 fliftf 5775 eloprabga 5937 ovmpt4g 5972 spc2ed 6209 eroveu 6600 supelti 6975 genpassl 7473 genpassu 7474 eqord1 8389 nn1suc 8884 bj-inex 13902 |
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