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Theorem fr2nr 5082
 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 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Proof of Theorem fr2nr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 4900 . . . . . . 7 {𝐵, 𝐶} ∈ V
21a1i 11 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ∈ V)
3 simpl 473 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝑅 Fr 𝐴)
4 prssi 4344 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
54adantl 482 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6 prnzg 4302 . . . . . . 7 (𝐵𝐴 → {𝐵, 𝐶} ≠ ∅)
76ad2antrl 763 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ≠ ∅)
8 fri 5066 . . . . . 6 ((({𝐵, 𝐶} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶} ⊆ 𝐴 ∧ {𝐵, 𝐶} ≠ ∅)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
92, 3, 5, 7, 8syl22anc 1325 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10 breq2 4648 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 308 . . . . . . . 8 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 2983 . . . . . . 7 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13 breq2 4648 . . . . . . . . 9 (𝑦 = 𝐶 → (𝑥𝑅𝑦𝑥𝑅𝐶))
1413notbid 308 . . . . . . . 8 (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
1514ralbidv 2983 . . . . . . 7 (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
1612, 15rexprg 4226 . . . . . 6 ((𝐵𝐴𝐶𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
1716adantl 482 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
189, 17mpbid 222 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19 prid2g 4287 . . . . . . 7 (𝐶𝐴𝐶 ∈ {𝐵, 𝐶})
2019ad2antll 764 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21 breq1 4647 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝑅𝐵𝐶𝑅𝐵))
2221notbid 308 . . . . . . 7 (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
2322rspcv 3300 . . . . . 6 (𝐶 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
2420, 23syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
25 prid1g 4286 . . . . . . 7 (𝐵𝐴𝐵 ∈ {𝐵, 𝐶})
2625ad2antrl 763 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27 breq1 4647 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
2827notbid 308 . . . . . . 7 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
2928rspcv 3300 . . . . . 6 (𝐵 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3026, 29syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3124, 30orim12d 882 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶) → (¬ 𝐶𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶)))
3218, 31mpd 15 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐶𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶))
3332orcomd 403 . 2 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34 ianor 509 . 2 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
3533, 34sylibr 224 1 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  ∃wrex 2910  Vcvv 3195   ⊆ wss 3567  ∅c0 3907  {cpr 4170   class class class wbr 4644   Fr wfr 5060 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-fr 5063 This theorem is referenced by:  efrn2lp  5086  dfwe2  6966
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