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Mirrors > Home > ILE Home > Th. List > 2oconcl | GIF version |
Description: Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
2oconcl | ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3593 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | difeq2 3229 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅)) | |
3 | dif0 3474 | . . . . . . . 8 ⊢ (1o ∖ ∅) = 1o | |
4 | 2, 3 | eqtrdi 2213 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o) |
5 | difeq2 3229 | . . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o)) | |
6 | difid 3472 | . . . . . . . 8 ⊢ (1o ∖ 1o) = ∅ | |
7 | 5, 6 | eqtrdi 2213 | . . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅) |
8 | 4, 7 | orim12i 749 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅)) |
9 | 8 | orcomd 719 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
10 | 1, 9 | syl 14 | . . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
11 | 1on 6382 | . . . . . 6 ⊢ 1o ∈ On | |
12 | difexg 4117 | . . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (1o ∖ 𝐴) ∈ V |
14 | 13 | elpr 3591 | . . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
15 | 10, 14 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o}) |
16 | df2o3 6389 | . . 3 ⊢ 2o = {∅, 1o} | |
17 | 15, 16 | eleqtrrdi 2258 | . 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o) |
18 | 17, 16 | eleq2s 2259 | 1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∖ cdif 3108 ∅c0 3404 {cpr 3571 Oncon0 4335 1oc1o 6368 2oc2o 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 df-1o 6375 df-2o 6376 |
This theorem is referenced by: ismkvnex 7110 |
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