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| 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 4648 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
| 2 | difeq2 4119 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅)) | |
| 3 | dif0 4377 | . . . . . . . 8 ⊢ (1o ∖ ∅) = 1o | |
| 4 | 2, 3 | eqtrdi 2792 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o) | 
| 5 | difeq2 4119 | . . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o)) | |
| 6 | difid 4375 | . . . . . . . 8 ⊢ (1o ∖ 1o) = ∅ | |
| 7 | 5, 6 | eqtrdi 2792 | . . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅) | 
| 8 | 4, 7 | orim12i 908 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅)) | 
| 9 | 8 | orcomd 871 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) | 
| 10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) | 
| 11 | 1on 8519 | . . . . . 6 ⊢ 1o ∈ On | |
| 12 | difexg 5328 | . . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V) | |
| 13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (1o ∖ 𝐴) ∈ V | 
| 14 | 13 | elpr 4649 | . . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) | 
| 15 | 10, 14 | sylibr 234 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o}) | 
| 16 | df2o3 8515 | . . 3 ⊢ 2o = {∅, 1o} | |
| 17 | 15, 16 | eleqtrrdi 2851 | . 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o) | 
| 18 | 17, 16 | eleq2s 2858 | 1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ∅c0 4332 {cpr 4627 Oncon0 6383 1oc1o 8500 2oc2o 8501 | 
| 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 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 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-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-suc 6389 df-1o 8507 df-2o 8508 | 
| This theorem is referenced by: efgmf 19732 efgmnvl 19733 efglem 19735 frgpuplem 19791 | 
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