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Theorem onssneli 6293
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 3958 . . 3 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
2 on.1 . . . . 5 𝐴 ∈ On
32oneli 6291 . . . 4 (𝐵𝐴𝐵 ∈ On)
4 eloni 6194 . . . 4 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 6202 . . . 4 (Ord 𝐵 → ¬ 𝐵𝐵)
63, 4, 53syl 18 . . 3 (𝐵𝐴 → ¬ 𝐵𝐵)
71, 6nsyli 160 . 2 (𝐴𝐵 → (𝐵𝐴 → ¬ 𝐵𝐴))
87pm2.01d 191 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2105  wss 3933  Ord word 6183  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  onsucconni  33682
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