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Theorem 2oconcl 7543
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 4175 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 difeq2 3706 . . . . . . . 8 (𝐴 = ∅ → (1𝑜𝐴) = (1𝑜 ∖ ∅))
3 dif0 3930 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2671 . . . . . . 7 (𝐴 = ∅ → (1𝑜𝐴) = 1𝑜)
5 difeq2 3706 . . . . . . . 8 (𝐴 = 1𝑜 → (1𝑜𝐴) = (1𝑜 ∖ 1𝑜))
6 difid 3928 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2671 . . . . . . 7 (𝐴 = 1𝑜 → (1𝑜𝐴) = ∅)
84, 7orim12i 538 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = 1𝑜 ∨ (1𝑜𝐴) = ∅))
98orcomd 403 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
101, 9syl 17 . . . 4 (𝐴 ∈ {∅, 1𝑜} → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
11 1on 7527 . . . . . 6 1𝑜 ∈ On
12 difexg 4778 . . . . . 6 (1𝑜 ∈ On → (1𝑜𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1𝑜𝐴) ∈ V
1413elpr 4176 . . . 4 ((1𝑜𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
1510, 14sylibr 224 . . 3 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ {∅, 1𝑜})
16 df2o3 7533 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2709 . 2 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ 2𝑜)
1817, 16eleq2s 2716 1 (𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  c0 3897  {cpr 4157  Oncon0 5692  1𝑜c1o 7513  2𝑜c2o 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-suc 5698  df-1o 7520  df-2o 7521
This theorem is referenced by:  efgmf  18066  efgmnvl  18067  efglem  18069  frgpuplem  18125
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