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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 38284, extssr 38510 and the comment of df-ssr 38499. (Contributed by Peter Mazsa, 5-Jul-2019.) |
| Ref | Expression |
|---|---|
| extid | ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvi 6161 | . . . . 5 ⊢ ◡ I = I | |
| 2 | 1 | eceq2i 8787 | . . . 4 ⊢ [𝐴]◡ I = [𝐴] I |
| 3 | ecidsn 8800 | . . . 4 ⊢ [𝐴] I = {𝐴} | |
| 4 | 2, 3 | eqtri 2765 | . . 3 ⊢ [𝐴]◡ I = {𝐴} |
| 5 | 1 | eceq2i 8787 | . . . 4 ⊢ [𝐵]◡ I = [𝐵] I |
| 6 | ecidsn 8800 | . . . 4 ⊢ [𝐵] I = {𝐵} | |
| 7 | 5, 6 | eqtri 2765 | . . 3 ⊢ [𝐵]◡ I = {𝐵} |
| 8 | 4, 7 | eqeq12i 2755 | . 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵}) |
| 9 | sneqbg 4843 | . 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 206 = wceq 1540 ∈ wcel 2108 {csn 4626 I cid 5577 ◡ccnv 5684 [cec 8743 |
| 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-in 3958 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 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 |
| This theorem is referenced by: (None) |
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