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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coss0 | Structured version Visualization version GIF version |
Description: Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.) |
Ref | Expression |
---|---|
coss0 | ⊢ ≀ ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcoss2 35821 | . 2 ⊢ ≀ ∅ = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} | |
2 | ec0 35781 | . . . . . . 7 ⊢ [𝑥]∅ = ∅ | |
3 | 2 | eleq2i 2881 | . . . . . 6 ⊢ (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅) |
4 | 2 | eleq2i 2881 | . . . . . 6 ⊢ (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅) |
5 | 3, 4 | anbi12i 629 | . . . . 5 ⊢ ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
6 | 5 | exbii 1849 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
7 | 19.9v 1988 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | |
8 | 6, 7 | bitri 278 | . . 3 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
9 | 8 | opabbii 5097 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} |
10 | prnzg 4674 | . . . . . 6 ⊢ (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅) | |
11 | 10 | elv 3446 | . . . . 5 ⊢ {𝑦, 𝑧} ≠ ∅ |
12 | ss0b 4305 | . . . . 5 ⊢ ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅) | |
13 | 11, 12 | nemtbir 3082 | . . . 4 ⊢ ¬ {𝑦, 𝑧} ⊆ ∅ |
14 | prssg 4712 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)) | |
15 | 14 | el2v 3448 | . . . 4 ⊢ ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅) |
16 | 13, 15 | mtbir 326 | . . 3 ⊢ ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) |
17 | 16 | opabf 35780 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅ |
18 | 1, 9, 17 | 3eqtri 2825 | 1 ⊢ ≀ ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 {cpr 4527 {copab 5092 [cec 8270 ≀ ccoss 35613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 df-coss 35819 |
This theorem is referenced by: (None) |
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