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Theorem onssneli 5980
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 3746 . . 3 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
2 on.1 . . . . 5 𝐴 ∈ On
32oneli 5978 . . . 4 (𝐵𝐴𝐵 ∈ On)
4 eloni 5876 . . . 4 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 5884 . . . 4 (Ord 𝐵 → ¬ 𝐵𝐵)
63, 4, 53syl 18 . . 3 (𝐵𝐴 → ¬ 𝐵𝐵)
71, 6nsyli 156 . 2 (𝐴𝐵 → (𝐵𝐴 → ¬ 𝐵𝐴))
87pm2.01d 181 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2145  wss 3723  Ord word 5865  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  onsucconni  32773
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