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Theorem ordnbtwnOLD 5817

Description: Obsolete proof of ordnbtwn 5816 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 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

**Proof of Theorem ordnbtwnOLD**
Step	Hyp	Ref	Expression
1		ordn2lp 5743	. . 3 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
2		ordirr 5741	. . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
3		ioran 511	. . 3 ⊢ (¬ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴) ↔ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∧ ¬ 𝐴 ∈ 𝐴))
4	1, 2, 3	sylanbrc 698	. 2 ⊢ (Ord 𝐴 → ¬ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
5		elsuci 5791	. . . . 5 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
6	5	anim2i 593	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
7		andi 911	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)))
8	6, 7	sylib 208	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)))
9		eleq2 2690	. . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐴))
10	9	biimpac 503	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → 𝐴 ∈ 𝐴)
11	10	orim2i 540	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ (𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴)) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
12	8, 11	syl 17	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐴))
13	4, 12	nsyl 135	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Ord word 5722 suc csuc 5725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-fr 5073 df-we 5075 df-ord 5726 df-suc 5729

This theorem is referenced by: (None)

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