Theorem fr2nr 5529

Description: A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
fr2nr	⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Proof of Theorem fr2nr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prex 5325	. . . . . . 7 ⊢ {𝐵, 𝐶} ∈ V
2	1	a1i 11	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ∈ V)
3		simpl 486	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		prssi 4734	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
5	4	adantl 485	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6		prnzg 4694	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵, 𝐶} ≠ ∅)
7	6	ad2antrl 728	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ≠ ∅)
8		fri 5512	. . . . . 6 ⊢ ((({𝐵, 𝐶} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶} ⊆ 𝐴 ∧ {𝐵, 𝐶} ≠ ∅)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
9	2, 3, 5, 7, 8	syl22anc 839	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10		breq2 5057	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 3118	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13		breq2 5057	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐶))
14	13	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
15	14	ralbidv 3118	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
16	12, 15	rexprg 4612	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
17	16	adantl 485	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
18	9, 17	mpbid 235	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19		prid2g 4677	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐵, 𝐶})
20	19	ad2antll 729	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21		breq1 5056	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝐵 ↔ 𝐶𝑅𝐵))
22	21	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
23	22	rspcv 3532	. . . . . 6 ⊢ (𝐶 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
24	20, 23	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
25		prid1g 4676	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵, 𝐶})
26	25	ad2antrl 728	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27		breq1 5056	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
28	27	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
29	28	rspcv 3532	. . . . . 6 ⊢ (𝐵 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
30	26, 29	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
31	24, 30	orim12d 965	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶) → (¬ 𝐶𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶)))
32	18, 31	mpd 15	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐶𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶))
33	32	orcomd 871	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34		ianor 982	. 2 ⊢ (¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
35	33, 34	sylibr 237	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ⊆ wss 3866  ∅c0 4237  {cpr 4543   class class class wbr 5053   Fr wfr 5506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-fr 5509

This theorem is referenced by:  efrn2lp  5533  dfwe2  7559