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Theorem onssneli 5796
Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onssneli (𝐴𝐵 → ¬ 𝐵𝐴)

Proof of Theorem onssneli
StepHypRef Expression
1 ssel 3577 . . 3 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
2 on.1 . . . . 5 𝐴 ∈ On
32oneli 5794 . . . 4 (𝐵𝐴𝐵 ∈ On)
4 eloni 5692 . . . 4 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 5700 . . . 4 (Ord 𝐵 → ¬ 𝐵𝐵)
63, 4, 53syl 18 . . 3 (𝐵𝐴 → ¬ 𝐵𝐵)
71, 6nsyli 155 . 2 (𝐴𝐵 → (𝐵𝐴 → ¬ 𝐵𝐴))
87pm2.01d 181 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1987  wss 3555  Ord word 5681  Oncon0 5682
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686
This theorem is referenced by:  onsucconni  32078
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