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Mirrors > Home > MPE Home > Th. List > Mathboxes > extid | Structured version Visualization version GIF version |
Description: Property of identity relation, see also extep 36789, extssr 37017 and the comment of df-ssr 37006. (Contributed by Peter Mazsa, 5-Jul-2019.) |
Ref | Expression |
---|---|
extid | ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvi 6095 | . . . . 5 ⊢ ◡ I = I | |
2 | 1 | eceq2i 8692 | . . . 4 ⊢ [𝐴]◡ I = [𝐴] I |
3 | ecidsn 8704 | . . . 4 ⊢ [𝐴] I = {𝐴} | |
4 | 2, 3 | eqtri 2761 | . . 3 ⊢ [𝐴]◡ I = {𝐴} |
5 | 1 | eceq2i 8692 | . . . 4 ⊢ [𝐵]◡ I = [𝐵] I |
6 | ecidsn 8704 | . . . 4 ⊢ [𝐵] I = {𝐵} | |
7 | 5, 6 | eqtri 2761 | . . 3 ⊢ [𝐵]◡ I = {𝐵} |
8 | 4, 7 | eqeq12i 2751 | . 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵}) |
9 | sneqbg 4802 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) | |
10 | 8, 9 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {csn 4587 I cid 5531 ◡ccnv 5633 [cec 8649 |
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 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8653 |
This theorem is referenced by: (None) |
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