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| Mirrors > Home > MPE Home > Th. List > issetid | Structured version Visualization version 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 5864 | . 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴) | |
| 2 | reli 5836 | . . 3 ⊢ Rel I | |
| 3 | 2 | brrelex1i 5741 | . 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V) | 
| 4 | 1, 3 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 I cid 5577 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: (None) | 
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