| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cossid | Structured version Visualization version GIF version | ||
| Description: Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.) |
| Ref | Expression |
|---|---|
| cossid | ⊢ ≀ I = I |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equvinv 2028 | . . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) | |
| 2 | ideqg 5860 | . . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
| 3 | 2 | elv 3484 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
| 4 | ideqg 5860 | . . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)) | |
| 5 | 4 | elv 3484 | . . . . . 6 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧) |
| 6 | 3, 5 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) |
| 7 | 6 | exbii 1848 | . . . 4 ⊢ (∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) |
| 8 | 1, 7 | bitr4i 278 | . . 3 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)) |
| 9 | 8 | opabbii 5208 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} |
| 10 | df-id 5576 | . 2 ⊢ I = {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} | |
| 11 | df-coss 38405 | . 2 ⊢ ≀ I = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} | |
| 12 | 9, 10, 11 | 3eqtr4ri 2775 | 1 ⊢ ≀ I = I |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 Vcvv 3479 class class class wbr 5141 {copab 5203 I cid 5575 ≀ ccoss 38175 |
| 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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| 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 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 df-id 5576 df-xp 5689 df-rel 5690 df-coss 38405 |
| This theorem is referenced by: cosscnvid 38475 eqvrelid 38783 |
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