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Theorem onssneli 6492
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 3973 . . 3 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
2 on.1 . . . . 5 𝐴 ∈ On
32oneli 6490 . . . 4 (𝐵𝐴𝐵 ∈ On)
4 eloni 6386 . . . 4 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 6394 . . . 4 (Ord 𝐵 → ¬ 𝐵𝐵)
63, 4, 53syl 18 . . 3 (𝐵𝐴 → ¬ 𝐵𝐵)
71, 6nsyli 157 . 2 (𝐴𝐵 → (𝐵𝐴 → ¬ 𝐵𝐴))
87pm2.01d 189 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2099  wss 3947  Ord word 6375  Oncon0 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380
This theorem is referenced by:  cofcutr  27941  onsucconni  36149
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