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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 4579 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | difeq2 4090 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅)) | |
3 | dif0 4329 | . . . . . . . 8 ⊢ (1o ∖ ∅) = 1o | |
4 | 2, 3 | syl6eq 2869 | . . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o) |
5 | difeq2 4090 | . . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o)) | |
6 | difid 4327 | . . . . . . . 8 ⊢ (1o ∖ 1o) = ∅ | |
7 | 5, 6 | syl6eq 2869 | . . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅) |
8 | 4, 7 | orim12i 902 | . . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅)) |
9 | 8 | orcomd 865 | . . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
11 | 1on 8098 | . . . . . 6 ⊢ 1o ∈ On | |
12 | difexg 5222 | . . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (1o ∖ 𝐴) ∈ V |
14 | 13 | elpr 4580 | . . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o)) |
15 | 10, 14 | sylibr 235 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o}) |
16 | df2o3 8106 | . . 3 ⊢ 2o = {∅, 1o} | |
17 | 15, 16 | eleqtrrdi 2921 | . 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o) |
18 | 17, 16 | eleq2s 2928 | 1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ∅c0 4288 {cpr 4559 Oncon0 6184 1oc1o 8084 2oc2o 8085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-suc 6190 df-1o 8091 df-2o 8092 |
This theorem is referenced by: efgmf 18768 efgmnvl 18769 efglem 18771 frgpuplem 18827 |
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