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Theorem ondif2 7534
Description: Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
ondif2 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))

Proof of Theorem ondif2
StepHypRef Expression
1 eldif 3569 . 2 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜))
2 1on 7519 . . . . 5 1𝑜 ∈ On
3 ontri1 5721 . . . . . 6 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ ¬ 1𝑜𝐴))
4 onsssuc 5777 . . . . . . 7 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜𝐴 ∈ suc 1𝑜))
5 df-2o 7513 . . . . . . . 8 2𝑜 = suc 1𝑜
65eleq2i 2690 . . . . . . 7 (𝐴 ∈ 2𝑜𝐴 ∈ suc 1𝑜)
74, 6syl6bbr 278 . . . . . 6 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜𝐴 ∈ 2𝑜))
83, 7bitr3d 270 . . . . 5 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (¬ 1𝑜𝐴𝐴 ∈ 2𝑜))
92, 8mpan2 706 . . . 4 (𝐴 ∈ On → (¬ 1𝑜𝐴𝐴 ∈ 2𝑜))
109con1bid 345 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2𝑜 ↔ 1𝑜𝐴))
1110pm5.32i 668 . 2 ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
121, 11bitri 264 1 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  wcel 1987  cdif 3556  wss 3559  Oncon0 5687  suc csuc 5689  1𝑜c1o 7505  2𝑜c2o 7506
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 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-suc 5693  df-1o 7512  df-2o 7513
This theorem is referenced by:  dif20el  7537  oeordi  7619  oewordi  7623  oaabs2  7677  omabs  7679  cnfcom3clem  8553  infxpenc2lem1  8793
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