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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 3469 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜)) | |
2 | difeq2 3112 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ ∅)) | |
3 | dif0 3353 | . . . . . . . 8 ⊢ (1𝑜 ∖ ∅) = 1𝑜 | |
4 | 2, 3 | syl6eq 2136 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = 1𝑜) |
5 | difeq2 3112 | . . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ 1𝑜)) | |
6 | difid 3351 | . . . . . . . 8 ⊢ (1𝑜 ∖ 1𝑜) = ∅ | |
7 | 5, 6 | syl6eq 2136 | . . . . . . 7 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = ∅) |
8 | 4, 7 | orim12i 711 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = 1𝑜 ∨ (1𝑜 ∖ 𝐴) = ∅)) |
9 | 8 | orcomd 683 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜)) |
10 | 1, 9 | syl 14 | . . . 4 ⊢ (𝐴 ∈ {∅, 1𝑜} → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜)) |
11 | 1on 6188 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
12 | difexg 3980 | . . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝐴) ∈ V) | |
13 | 11, 12 | ax-mp 7 | . . . . 5 ⊢ (1𝑜 ∖ 𝐴) ∈ V |
14 | 13 | elpr 3467 | . . . 4 ⊢ ((1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜)) |
15 | 10, 14 | sylibr 132 | . . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜}) |
16 | df2o3 6195 | . . 3 ⊢ 2𝑜 = {∅, 1𝑜} | |
17 | 15, 16 | syl6eleqr 2181 | . 2 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ 2𝑜) |
18 | 17, 16 | eleq2s 2182 | 1 ⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 664 = wceq 1289 ∈ wcel 1438 Vcvv 2619 ∖ cdif 2996 ∅c0 3286 {cpr 3447 Oncon0 4190 1𝑜c1o 6174 2𝑜c2o 6175 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-tr 3937 df-iord 4193 df-on 4195 df-suc 4198 df-1o 6181 df-2o 6182 |
This theorem is referenced by: (None) |
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