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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 2025 | . . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) | |
2 | ideqg 5864 | . . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
3 | 2 | elv 3482 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | ideqg 5864 | . . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)) | |
5 | 4 | elv 3482 | . . . . . 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 1844 | . . . 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 5214 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} |
10 | df-id 5582 | . 2 ⊢ I = {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} | |
11 | df-coss 38392 | . 2 ⊢ ≀ I = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} | |
12 | 9, 10, 11 | 3eqtr4ri 2773 | 1 ⊢ ≀ I = I |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 Vcvv 3477 class class class wbr 5147 {copab 5209 I cid 5581 ≀ ccoss 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-coss 38392 |
This theorem is referenced by: cosscnvid 38462 eqvrelid 38770 |
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