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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 37885 | . 2 ⊢ ≀ ∅ = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} | |
2 | ec0 37841 | . . . . . . 7 ⊢ [𝑥]∅ = ∅ | |
3 | 2 | eleq2i 2821 | . . . . . 6 ⊢ (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅) |
4 | 2 | eleq2i 2821 | . . . . . 6 ⊢ (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅) |
5 | 3, 4 | anbi12i 627 | . . . . 5 ⊢ ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
6 | 5 | exbii 1843 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
7 | 19.9v 1980 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | |
8 | 6, 7 | bitri 275 | . . 3 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
9 | 8 | opabbii 5215 | . 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} |
10 | prnzg 4783 | . . . . . 6 ⊢ (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅) | |
11 | 10 | elv 3477 | . . . . 5 ⊢ {𝑦, 𝑧} ≠ ∅ |
12 | ss0b 4398 | . . . . 5 ⊢ ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅) | |
13 | 11, 12 | nemtbir 3035 | . . . 4 ⊢ ¬ {𝑦, 𝑧} ⊆ ∅ |
14 | prssg 4823 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)) | |
15 | 14 | el2v 3479 | . . . 4 ⊢ ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅) |
16 | 13, 15 | mtbir 323 | . . 3 ⊢ ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) |
17 | 16 | opabf 37840 | . 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅ |
18 | 1, 9, 17 | 3eqtri 2760 | 1 ⊢ ≀ ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 {cpr 4631 {copab 5210 [cec 8722 ≀ ccoss 37648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8726 df-coss 37883 |
This theorem is referenced by: eqvrel0 38258 |
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