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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemelo | Structured version Visualization version GIF version | ||
| Description: Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.) | 
| Ref | Expression | 
|---|---|
| ballotth.m | ⊢ 𝑀 ∈ ℕ | 
| ballotth.n | ⊢ 𝑁 ∈ ℕ | 
| ballotth.o | ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | 
| Ref | Expression | 
|---|---|
| ballotlemelo | ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveqeq2 6914 | . . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀)) | |
| 2 | ballotth.o | . . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
| 3 | fveqeq2 6914 | . . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀)) | |
| 4 | 3 | cbvrabv 3446 | . . . 4 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀} | 
| 5 | 2, 4 | eqtri 2764 | . . 3 ⊢ 𝑂 = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀} | 
| 6 | 1, 5 | elrab2 3694 | . 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) | 
| 7 | ovex 7465 | . . . 4 ⊢ (1...(𝑀 + 𝑁)) ∈ V | |
| 8 | 7 | elpw2 5333 | . . 3 ⊢ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝐶 ⊆ (1...(𝑀 + 𝑁))) | 
| 9 | 8 | anbi1i 624 | . 2 ⊢ ((𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) | 
| 10 | 6, 9 | bitri 275 | 1 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 𝒫 cpw 4599 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 ℕcn 12267 ...cfz 13548 ♯chash 14370 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: ballotlemscr 34522 ballotlemro 34526 ballotlemfg 34529 ballotlemfrc 34530 ballotlemfrceq 34532 ballotlemrinv0 34536 | 
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