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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 34984, extssr 35194 and the comment of df-ssr 35183. (Contributed by Peter Mazsa, 5-Jul-2019.) |
Ref | Expression |
---|---|
extid | ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvi 5834 | . . . . 5 ⊢ ◡ I = I | |
2 | 1 | eceq2i 8124 | . . . 4 ⊢ [𝐴]◡ I = [𝐴] I |
3 | ecidsn 8136 | . . . 4 ⊢ [𝐴] I = {𝐴} | |
4 | 2, 3 | eqtri 2796 | . . 3 ⊢ [𝐴]◡ I = {𝐴} |
5 | 1 | eceq2i 8124 | . . . 4 ⊢ [𝐵]◡ I = [𝐵] I |
6 | ecidsn 8136 | . . . 4 ⊢ [𝐵] I = {𝐵} | |
7 | 5, 6 | eqtri 2796 | . . 3 ⊢ [𝐵]◡ I = {𝐵} |
8 | 4, 7 | eqeq12i 2786 | . 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵}) |
9 | sneqbg 4642 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) | |
10 | 8, 9 | syl5bb 275 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 {csn 4435 I cid 5305 ◡ccnv 5400 [cec 8081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-ec 8085 |
This theorem is referenced by: (None) |
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