Theorem fr2nr 5016

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 4836	. . . . . . 7 ⊢ {𝐵, 𝐶} ∈ V
2	1	a1i 11	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ∈ V)
3		simpl 472	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		prssi 4293	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
5	4	adantl 481	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6		prnzg 4254	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵, 𝐶} ≠ ∅)
7	6	ad2antrl 760	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ≠ ∅)
8		fri 5000	. . . . . 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 1319	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10		breq2 4587	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 307	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 2969	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13		breq2 4587	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐶))
14	13	notbid 307	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
15	14	ralbidv 2969	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
16	12, 15	rexprg 4182	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
17	16	adantl 481	. . . . 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 221	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19		prid2g 4240	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐵, 𝐶})
20	19	ad2antll 761	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21		breq1 4586	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝐵 ↔ 𝐶𝑅𝐵))
22	21	notbid 307	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
23	22	rspcv 3278	. . . . . 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 4239	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵, 𝐶})
26	25	ad2antrl 760	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27		breq1 4586	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
28	27	notbid 307	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
29	28	rspcv 3278	. . . . . 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 879	. . . 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 402	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34		ianor 508	. 2 ⊢ (¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
35	33, 34	sylibr 223	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {cpr 4127   class class class wbr 4583   Fr wfr 4994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-fr 4997

This theorem is referenced by:  efrn2lp  5020  dfwe2  6873