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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 3550 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | difeq2 3188 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅)) | |
3 | dif0 3433 | . . . . . . . 8 ⊢ (1o ∖ ∅) = 1o | |
4 | 2, 3 | syl6eq 2188 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o) |
5 | difeq2 3188 | . . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o)) | |
6 | difid 3431 | . . . . . . . 8 ⊢ (1o ∖ 1o) = ∅ | |
7 | 5, 6 | syl6eq 2188 | . . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅) |
8 | 4, 7 | orim12i 748 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅)) |
9 | 8 | orcomd 718 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
10 | 1, 9 | syl 14 | . . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
11 | 1on 6320 | . . . . . 6 ⊢ 1o ∈ On | |
12 | difexg 4069 | . . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (1o ∖ 𝐴) ∈ V |
14 | 13 | elpr 3548 | . . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
15 | 10, 14 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o}) |
16 | df2o3 6327 | . . 3 ⊢ 2o = {∅, 1o} | |
17 | 15, 16 | eleqtrrdi 2233 | . 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o) |
18 | 17, 16 | eleq2s 2234 | 1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ∅c0 3363 {cpr 3528 Oncon0 4285 1oc1o 6306 2oc2o 6307 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 df-1o 6313 df-2o 6314 |
This theorem is referenced by: ismkvnex 7029 |
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