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Theorem fr2nr 5616
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 5394 . . . . . . 7 {𝐵, 𝐶} ∈ V
21a1i 11 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ∈ V)
3 simpl 484 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝑅 Fr 𝐴)
4 prssi 4786 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
54adantl 483 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6 prnzg 4744 . . . . . . 7 (𝐵𝐴 → {𝐵, 𝐶} ≠ ∅)
76ad2antrl 727 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ≠ ∅)
8 fri 5598 . . . . . 6 ((({𝐵, 𝐶} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶} ⊆ 𝐴 ∧ {𝐵, 𝐶} ≠ ∅)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
92, 3, 5, 7, 8syl22anc 838 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10 breq2 5114 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 318 . . . . . . . 8 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 3175 . . . . . . 7 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13 breq2 5114 . . . . . . . . 9 (𝑦 = 𝐶 → (𝑥𝑅𝑦𝑥𝑅𝐶))
1413notbid 318 . . . . . . . 8 (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
1514ralbidv 3175 . . . . . . 7 (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
1612, 15rexprg 4662 . . . . . 6 ((𝐵𝐴𝐶𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
1716adantl 483 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
189, 17mpbid 231 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19 prid2g 4727 . . . . . . 7 (𝐶𝐴𝐶 ∈ {𝐵, 𝐶})
2019ad2antll 728 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21 breq1 5113 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝑅𝐵𝐶𝑅𝐵))
2221notbid 318 . . . . . . 7 (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
2322rspcv 3580 . . . . . 6 (𝐶 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
2420, 23syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
25 prid1g 4726 . . . . . . 7 (𝐵𝐴𝐵 ∈ {𝐵, 𝐶})
2625ad2antrl 727 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27 breq1 5113 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
2827notbid 318 . . . . . . 7 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
2928rspcv 3580 . . . . . 6 (𝐵 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3026, 29syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3124, 30orim12d 964 . . . 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 870 . 2 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34 ianor 981 . 2 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
3533, 34sylibr 233 1 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  wss 3915  c0 4287  {cpr 4593   class class class wbr 5110   Fr wfr 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-fr 5593
This theorem is referenced by:  efrn2lp  5620  dfwe2  7713
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