Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 2035 | . . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) | |
2 | ideqg 5725 | . . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
3 | 2 | elv 3502 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | ideqg 5725 | . . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)) | |
5 | 4 | elv 3502 | . . . . . 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 1847 | . . . 4 ⊢ (∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) |
8 | 1, 7 | bitr4i 280 | . . 3 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)) |
9 | 8 | opabbii 5136 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} |
10 | df-id 5463 | . 2 ⊢ I = {〈𝑦, 𝑧〉 ∣ 𝑦 = 𝑧} | |
11 | df-coss 35663 | . 2 ⊢ ≀ I = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} | |
12 | 9, 10, 11 | 3eqtr4ri 2858 | 1 ⊢ ≀ I = I |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 Vcvv 3497 class class class wbr 5069 {copab 5131 I cid 5462 ≀ ccoss 35457 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-coss 35663 |
This theorem is referenced by: cosscnvid 35725 |
Copyright terms: Public domain | W3C validator |