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Theorem ordnbtwnOLD 5855
 Description: Obsolete proof of ordnbtwn 5854 as of 24-Sep-2021. (Contributed by NM, 21-Jun-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ordnbtwnOLD (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Proof of Theorem ordnbtwnOLD
StepHypRef Expression
1 ordn2lp 5781 . . 3 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
2 ordirr 5779 . . 3 (Ord 𝐴 → ¬ 𝐴𝐴)
3 ioran 510 . . 3 (¬ ((𝐴𝐵𝐵𝐴) ∨ 𝐴𝐴) ↔ (¬ (𝐴𝐵𝐵𝐴) ∧ ¬ 𝐴𝐴))
41, 2, 3sylanbrc 699 . 2 (Ord 𝐴 → ¬ ((𝐴𝐵𝐵𝐴) ∨ 𝐴𝐴))
5 elsuci 5829 . . . . 5 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
65anim2i 592 . . . 4 ((𝐴𝐵𝐵 ∈ suc 𝐴) → (𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)))
7 andi 929 . . . 4 ((𝐴𝐵 ∧ (𝐵𝐴𝐵 = 𝐴)) ↔ ((𝐴𝐵𝐵𝐴) ∨ (𝐴𝐵𝐵 = 𝐴)))
86, 7sylib 208 . . 3 ((𝐴𝐵𝐵 ∈ suc 𝐴) → ((𝐴𝐵𝐵𝐴) ∨ (𝐴𝐵𝐵 = 𝐴)))
9 eleq2 2719 . . . . 5 (𝐵 = 𝐴 → (𝐴𝐵𝐴𝐴))
109biimpac 502 . . . 4 ((𝐴𝐵𝐵 = 𝐴) → 𝐴𝐴)
1110orim2i 539 . . 3 (((𝐴𝐵𝐵𝐴) ∨ (𝐴𝐵𝐵 = 𝐴)) → ((𝐴𝐵𝐵𝐴) ∨ 𝐴𝐴))
128, 11syl 17 . 2 ((𝐴𝐵𝐵 ∈ suc 𝐴) → ((𝐴𝐵𝐵𝐴) ∨ 𝐴𝐴))
134, 12nsyl 135 1 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Ord word 5760  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-fr 5102  df-we 5104  df-ord 5764  df-suc 5767 This theorem is referenced by: (None)
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