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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 35541 | . 2 ⊢ ≀ ∅ = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} | |
2 | ec0 35501 | . . . . . . 7 ⊢ [𝑥]∅ = ∅ | |
3 | 2 | eleq2i 2901 | . . . . . 6 ⊢ (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅) |
4 | 2 | eleq2i 2901 | . . . . . 6 ⊢ (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅) |
5 | 3, 4 | anbi12i 626 | . . . . 5 ⊢ ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
6 | 5 | exbii 1839 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
7 | 19.9v 1979 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | |
8 | 6, 7 | bitri 276 | . . 3 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
9 | 8 | opabbii 5124 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} |
10 | prnzg 4705 | . . . . . 6 ⊢ (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅) | |
11 | 10 | elv 3497 | . . . . 5 ⊢ {𝑦, 𝑧} ≠ ∅ |
12 | ss0b 4348 | . . . . 5 ⊢ ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅) | |
13 | 11, 12 | nemtbir 3109 | . . . 4 ⊢ ¬ {𝑦, 𝑧} ⊆ ∅ |
14 | prssg 4744 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)) | |
15 | 14 | el2v 3499 | . . . 4 ⊢ ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅) |
16 | 13, 15 | mtbir 324 | . . 3 ⊢ ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) |
17 | 16 | opabf 35500 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅ |
18 | 1, 9, 17 | 3eqtri 2845 | 1 ⊢ ≀ ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 {cpr 4559 {copab 5119 [cec 8276 ≀ ccoss 35334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 df-coss 35539 |
This theorem is referenced by: (None) |
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