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Theorem 2oconcl 5937
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 3370 . . . . 5 (A {∅, 1𝑜} → (A = ∅ A = 1𝑜))
2 difeq2 3033 . . . . . . . 8 (A = ∅ → (1𝑜A) = (1𝑜 ∖ ∅))
3 dif0 3271 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2070 . . . . . . 7 (A = ∅ → (1𝑜A) = 1𝑜)
5 difeq2 3033 . . . . . . . 8 (A = 1𝑜 → (1𝑜A) = (1𝑜 ∖ 1𝑜))
6 difid 3269 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2070 . . . . . . 7 (A = 1𝑜 → (1𝑜A) = ∅)
84, 7orim12i 663 . . . . . 6 ((A = ∅ A = 1𝑜) → ((1𝑜A) = 1𝑜 (1𝑜A) = ∅))
98orcomd 635 . . . . 5 ((A = ∅ A = 1𝑜) → ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
101, 9syl 14 . . . 4 (A {∅, 1𝑜} → ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
11 1on 5923 . . . . . 6 1𝑜 On
12 difexg 3872 . . . . . 6 (1𝑜 On → (1𝑜A) V)
1311, 12ax-mp 7 . . . . 5 (1𝑜A) V
1413elpr 3368 . . . 4 ((1𝑜A) {∅, 1𝑜} ↔ ((1𝑜A) = ∅ (1𝑜A) = 1𝑜))
1510, 14sylibr 137 . . 3 (A {∅, 1𝑜} → (1𝑜A) {∅, 1𝑜})
16 df2o3 5929 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2113 . 2 (A {∅, 1𝑜} → (1𝑜A) 2𝑜)
1817, 16eleq2s 2114 1 (A 2𝑜 → (1𝑜A) 2𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228   wcel 1374  Vcvv 2535  cdif 2891  c0 3201  {cpr 3351  Oncon0 4049  1𝑜c1o 5909  2𝑜c2o 5910
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-1o 5916  df-2o 5917
This theorem is referenced by: (None)
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