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Theorem 2oconcl 5961
Description: Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
2oconcl (A 2𝑜 → (1𝑜A) 2𝑜)

Proof of Theorem 2oconcl
StepHypRef Expression
1 elpri 3387 . . . . 5 (A {∅, 1𝑜} → (A = ∅ A = 1𝑜))
2 difeq2 3050 . . . . . . . 8 (A = ∅ → (1𝑜A) = (1𝑜 ∖ ∅))
3 dif0 3288 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2085 . . . . . . 7 (A = ∅ → (1𝑜A) = 1𝑜)
5 difeq2 3050 . . . . . . . 8 (A = 1𝑜 → (1𝑜A) = (1𝑜 ∖ 1𝑜))
6 difid 3286 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2085 . . . . . . 7 (A = 1𝑜 → (1𝑜A) = ∅)
84, 7orim12i 675 . . . . . 6 ((A = ∅ A = 1𝑜) → ((1𝑜A) = 1𝑜 (1𝑜A) = ∅))
98orcomd 647 . . . . 5 ((A = ∅ A = 1𝑜) → ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
101, 9syl 14 . . . 4 (A {∅, 1𝑜} → ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
11 1on 5947 . . . . . 6 1𝑜 On
12 difexg 3889 . . . . . 6 (1𝑜 On → (1𝑜A) V)
1311, 12ax-mp 7 . . . . 5 (1𝑜A) V
1413elpr 3385 . . . 4 ((1𝑜A) {∅, 1𝑜} ↔ ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
1510, 14sylibr 137 . . 3 (A {∅, 1𝑜} → (1𝑜A) {∅, 1𝑜})
16 df2o3 5953 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2128 . 2 (A {∅, 1𝑜} → (1𝑜A) 2𝑜)
1817, 16eleq2s 2129 1 (A 2𝑜 → (1𝑜A) 2𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  cdif 2908  c0 3218  {cpr 3368  Oncon0 4066  1𝑜c1o 5933  2𝑜c2o 5934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-1o 5940  df-2o 5941
This theorem is referenced by: (None)
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