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Theorem nndomog 8692
Description: Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 8697 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 8697. (Revised by RP, 5-Nov-2023.)
Assertion
Ref Expression
nndomog ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))

Proof of Theorem nndomog
StepHypRef Expression
1 php2 8686 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵𝐴)
21ex 416 . . . . 5 (𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
3 domnsym 8627 . . . . 5 (𝐴𝐵 → ¬ 𝐵𝐴)
42, 3nsyli 160 . . . 4 (𝐴 ∈ ω → (𝐴𝐵 → ¬ 𝐵𝐴))
54adantr 484 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵 → ¬ 𝐵𝐴))
6 nnord 7568 . . . 4 (𝐴 ∈ ω → Ord 𝐴)
7 eloni 6169 . . . 4 (𝐵 ∈ On → Ord 𝐵)
8 ordtri1 6192 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
9 ordelpss 6187 . . . . . . 7 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
109ancoms 462 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
1110notbid 321 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐴))
128, 11bitrd 282 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
136, 7, 12syl2an 598 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
145, 13sylibrd 262 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
15 ssdomg 8538 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
1615adantl 485 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
1714, 16impbid 215 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2111  wss 3881  wpss 3882   class class class wbr 5030  Ord word 6158  Oncon0 6159  ωcom 7560  cdom 8490  csdm 8491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495
This theorem is referenced by:  nndomo  8697  harsucnn  9411
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