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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 37786 | . 2 ⊢ ≀ ∅ = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} | |
2 | ec0 37741 | . . . . . . 7 ⊢ [𝑥]∅ = ∅ | |
3 | 2 | eleq2i 2817 | . . . . . 6 ⊢ (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅) |
4 | 2 | eleq2i 2817 | . . . . . 6 ⊢ (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅) |
5 | 3, 4 | anbi12i 626 | . . . . 5 ⊢ ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
6 | 5 | exbii 1842 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
7 | 19.9v 1979 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | |
8 | 6, 7 | bitri 275 | . . 3 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) |
9 | 8 | opabbii 5206 | . 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} |
10 | prnzg 4775 | . . . . . 6 ⊢ (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅) | |
11 | 10 | elv 3472 | . . . . 5 ⊢ {𝑦, 𝑧} ≠ ∅ |
12 | ss0b 4390 | . . . . 5 ⊢ ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅) | |
13 | 11, 12 | nemtbir 3030 | . . . 4 ⊢ ¬ {𝑦, 𝑧} ⊆ ∅ |
14 | prssg 4815 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)) | |
15 | 14 | el2v 3474 | . . . 4 ⊢ ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅) |
16 | 13, 15 | mtbir 323 | . . 3 ⊢ ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) |
17 | 16 | opabf 37740 | . 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅ |
18 | 1, 9, 17 | 3eqtri 2756 | 1 ⊢ ≀ ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ⊆ wss 3941 ∅c0 4315 {cpr 4623 {copab 5201 [cec 8698 ≀ ccoss 37546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 df-coss 37784 |
This theorem is referenced by: eqvrel0 38159 |
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