![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5810 | . 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴) | |
2 | reli 5783 | . . 3 ⊢ Rel I | |
3 | 2 | brrelex1i 5689 | . 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V) |
4 | 1, 3 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 Vcvv 3444 class class class wbr 5106 I cid 5531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |