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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 2668 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | isset 2663 | . 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 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-v 2659 |
This theorem is referenced by: elex22 2672 elex2 2673 ceqsalt 2683 ceqsalg 2685 cgsexg 2692 cgsex2g 2693 cgsex4g 2694 vtoclgft 2707 vtocleg 2728 vtoclegft 2729 spc2egv 2746 spc2gv 2747 spc3egv 2748 spc3gv 2749 eqvincg 2779 tpid3g 3604 iinexgm 4039 copsex2t 4127 copsex2g 4128 ralxfr2d 4345 rexxfr2d 4346 fliftf 5654 eloprabga 5812 ovmpt4g 5847 spc2ed 6084 eroveu 6474 supelti 6841 genpassl 7280 genpassu 7281 eqord1 8164 nn1suc 8649 bj-inex 12797 |
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