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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 36743, extssr 36971 and the comment of df-ssr 36960. (Contributed by Peter Mazsa, 5-Jul-2019.) |
Ref | Expression |
---|---|
extid | ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvi 6094 | . . . . 5 ⊢ ◡ I = I | |
2 | 1 | eceq2i 8689 | . . . 4 ⊢ [𝐴]◡ I = [𝐴] I |
3 | ecidsn 8701 | . . . 4 ⊢ [𝐴] I = {𝐴} | |
4 | 2, 3 | eqtri 2764 | . . 3 ⊢ [𝐴]◡ I = {𝐴} |
5 | 1 | eceq2i 8689 | . . . 4 ⊢ [𝐵]◡ I = [𝐵] I |
6 | ecidsn 8701 | . . . 4 ⊢ [𝐵] I = {𝐵} | |
7 | 5, 6 | eqtri 2764 | . . 3 ⊢ [𝐵]◡ I = {𝐵} |
8 | 4, 7 | eqeq12i 2754 | . 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵}) |
9 | sneqbg 4801 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) | |
10 | 8, 9 | bitrid 282 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 {csn 4586 I cid 5530 ◡ccnv 5632 [cec 8646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ec 8650 |
This theorem is referenced by: (None) |
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