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Theorem fr2nr 5002
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 4827 . . . . . . 7 {𝐵, 𝐶} ∈ V
21a1i 11 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ∈ V)
3 simpl 471 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝑅 Fr 𝐴)
4 prssi 4288 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
54adantl 480 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
6 prnzg 4249 . . . . . . 7 (𝐵𝐴 → {𝐵, 𝐶} ≠ ∅)
76ad2antrl 759 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → {𝐵, 𝐶} ≠ ∅)
8 fri 4986 . . . . . 6 ((({𝐵, 𝐶} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶} ⊆ 𝐴 ∧ {𝐵, 𝐶} ≠ ∅)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
92, 3, 5, 7, 8syl22anc 1318 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦)
10 breq2 4577 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 306 . . . . . . . 8 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 2964 . . . . . . 7 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵))
13 breq2 4577 . . . . . . . . 9 (𝑦 = 𝐶 → (𝑥𝑅𝑦𝑥𝑅𝐶))
1413notbid 306 . . . . . . . 8 (𝑦 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐶))
1514ralbidv 2964 . . . . . . 7 (𝑦 = 𝐶 → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
1612, 15rexprg 4177 . . . . . 6 ((𝐵𝐴𝐶𝐴) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
1716adantl 480 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∃𝑦 ∈ {𝐵, 𝐶}∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝑦 ↔ (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶)))
189, 17mpbid 220 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 ∨ ∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶))
19 prid2g 4235 . . . . . . 7 (𝐶𝐴𝐶 ∈ {𝐵, 𝐶})
2019ad2antll 760 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ {𝐵, 𝐶})
21 breq1 4576 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝑅𝐵𝐶𝑅𝐵))
2221notbid 306 . . . . . . 7 (𝑥 = 𝐶 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
2322rspcv 3273 . . . . . 6 (𝐶 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
2420, 23syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐵 → ¬ 𝐶𝑅𝐵))
25 prid1g 4234 . . . . . . 7 (𝐵𝐴𝐵 ∈ {𝐵, 𝐶})
2625ad2antrl 759 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ {𝐵, 𝐶})
27 breq1 4576 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
2827notbid 306 . . . . . . 7 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
2928rspcv 3273 . . . . . 6 (𝐵 ∈ {𝐵, 𝐶} → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3026, 29syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (∀𝑥 ∈ {𝐵, 𝐶} ¬ 𝑥𝑅𝐶 → ¬ 𝐵𝑅𝐶))
3124, 30orim12d 878 . . . 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 401 . 2 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
34 ianor 507 . 2 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐵))
3533, 34sylibr 222 1 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  Vcvv 3168  wss 3535  c0 3869  {cpr 4122   class class class wbr 4573   Fr wfr 4980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-fr 4983
This theorem is referenced by:  efrn2lp  5006  dfwe2  6846
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