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Mirrors > Home > ILE Home > Th. List > issetid | GIF version |
Description: Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
issetid | ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ididg 4779 | . 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴) | |
2 | reli 4755 | . . 3 ⊢ Rel I | |
3 | 2 | brrelex1i 4668 | . 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V) |
4 | 1, 3 | impbii 126 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 Vcvv 2737 class class class wbr 4002 I cid 4287 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 |
This theorem is referenced by: (None) |
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