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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 2033 | . . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧)) | |
2 | ideqg 5808 | . . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
3 | 2 | elv 3450 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | ideqg 5808 | . . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)) | |
5 | 4 | elv 3450 | . . . . . 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 1851 | . . . 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 5173 | . 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} |
10 | df-id 5532 | . 2 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} | |
11 | df-coss 36919 | . 2 ⊢ ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)} | |
12 | 9, 10, 11 | 3eqtr4ri 2772 | 1 ⊢ ≀ I = I |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 Vcvv 3444 class class class wbr 5106 {copab 5168 I cid 5531 ≀ ccoss 36680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-coss 36919 |
This theorem is referenced by: cosscnvid 36989 eqvrelid 37297 |
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