| Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ideq2 | Structured version Visualization version GIF version | ||
| Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.) |
| Ref | Expression |
|---|---|
| ideq2 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brid 38307 | . 2 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) | |
| 2 | ideqg 5862 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐵 = 𝐴)) | |
| 3 | eqcom 2744 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
| 4 | 2, 3 | bitrdi 287 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐴 = 𝐵)) |
| 5 | 1, 4 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 I cid 5577 |
| 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-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 |
| This theorem is referenced by: br1cossinidres 38450 br1cossxrnidres 38452 |
| Copyright terms: Public domain | W3C validator |