|   | Mathbox for Peter Mazsa | < Previous  
      Next > Nearby theorems | |
| 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 38414 | . 2 ⊢ ≀ ∅ = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} | |
| 2 | ec0 38370 | . . . . . . 7 ⊢ [𝑥]∅ = ∅ | |
| 3 | 2 | eleq2i 2833 | . . . . . 6 ⊢ (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅) | 
| 4 | 2 | eleq2i 2833 | . . . . . 6 ⊢ (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅) | 
| 5 | 3, 4 | anbi12i 628 | . . . . 5 ⊢ ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | 
| 6 | 5 | exbii 1848 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | 
| 7 | 19.9v 1983 | . . . 4 ⊢ (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | |
| 8 | 6, 7 | bitri 275 | . . 3 ⊢ (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)) | 
| 9 | 8 | opabbii 5210 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} | 
| 10 | prnzg 4778 | . . . . . 6 ⊢ (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅) | |
| 11 | 10 | elv 3485 | . . . . 5 ⊢ {𝑦, 𝑧} ≠ ∅ | 
| 12 | ss0b 4401 | . . . . 5 ⊢ ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅) | |
| 13 | 11, 12 | nemtbir 3038 | . . . 4 ⊢ ¬ {𝑦, 𝑧} ⊆ ∅ | 
| 14 | prssg 4819 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)) | |
| 15 | 14 | el2v 3487 | . . . 4 ⊢ ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅) | 
| 16 | 13, 15 | mtbir 323 | . . 3 ⊢ ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) | 
| 17 | 16 | opabf 38369 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅ | 
| 18 | 1, 9, 17 | 3eqtri 2769 | 1 ⊢ ≀ ∅ = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {cpr 4628 {copab 5205 [cec 8743 ≀ ccoss 38182 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-coss 38412 | 
| This theorem is referenced by: eqvrel0 38787 | 
| Copyright terms: Public domain | W3C validator |