Theorem fr2nr 5595

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 5367	. . . . . . 7 ⊢ {𝐵, 𝐶} ∈ V
2	1	a1i 11	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ∈ V)
3		simpl 483	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		prssi 4752	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
5	4	adantl 482	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6		prnzg 4710	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵, 𝐶} ≠ ∅)
7	6	ad2antrl 734	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → {𝐵, 𝐶} ≠ ∅)
8		fri 5576	. . . . . 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 844	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10		breq2 5076	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 319	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 3162	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13		breq2 5076	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐶))
14	13	notbid 319	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
15	14	ralbidv 3162	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
16	12, 15	rexprg 4629	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
17	16	adantl 482	. . . . 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 233	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19		prid2g 4693	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐵, 𝐶})
20	19	ad2antll 735	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21		breq1 5075	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝐵 ↔ 𝐶𝑅𝐵))
22	21	notbid 319	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
23	22	rspcv 3556	. . . . . 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 4692	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵, 𝐶})
26	25	ad2antrl 734	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27		breq1 5075	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
28	27	notbid 319	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
29	28	rspcv 3556	. . . . . 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 972	. . . 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 877	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34		ianor 989	. 2 ⊢ (¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
35	33, 34	sylibr 235	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  {cpr 4557   class class class wbr 5072   Fr wfr 5568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-fr 5571

This theorem is referenced by:  efrn2lp  5599  dfwe2  7717