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Mirrors > Home > MPE Home > Th. List > 2oconcl | Structured version Visualization version 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 4653 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | difeq2 4114 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅)) | |
3 | dif0 4374 | . . . . . . . 8 ⊢ (1o ∖ ∅) = 1o | |
4 | 2, 3 | eqtrdi 2783 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o) |
5 | difeq2 4114 | . . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o)) | |
6 | difid 4372 | . . . . . . . 8 ⊢ (1o ∖ 1o) = ∅ | |
7 | 5, 6 | eqtrdi 2783 | . . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅) |
8 | 4, 7 | orim12i 906 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅)) |
9 | 8 | orcomd 869 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
11 | 1on 8503 | . . . . . 6 ⊢ 1o ∈ On | |
12 | difexg 5331 | . . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (1o ∖ 𝐴) ∈ V |
14 | 13 | elpr 4654 | . . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
15 | 10, 14 | sylibr 233 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o}) |
16 | df2o3 8499 | . . 3 ⊢ 2o = {∅, 1o} | |
17 | 15, 16 | eleqtrrdi 2839 | . 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o) |
18 | 17, 16 | eleq2s 2846 | 1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ∖ cdif 3944 ∅c0 4324 {cpr 4632 Oncon0 6372 1oc1o 8484 2oc2o 8485 |
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-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-tr 5268 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-ord 6375 df-on 6376 df-suc 6378 df-1o 8491 df-2o 8492 |
This theorem is referenced by: efgmf 19673 efgmnvl 19674 efglem 19676 frgpuplem 19732 |
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