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Theorem tratrb 41162
Description: If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 41487. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tratrb ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem tratrb
StepHypRef Expression
1 nfv 1916 . . . 4 𝑥Tr 𝐴
2 nfra1 3213 . . . 4 𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
3 nfv 1916 . . . 4 𝑥 𝐵𝐴
41, 2, 3nf3an 1903 . . 3 𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
5 nfv 1916 . . . . 5 𝑦Tr 𝐴
6 nfra2w 3221 . . . . 5 𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
7 nfv 1916 . . . . 5 𝑦 𝐵𝐴
85, 6, 7nf3an 1903 . . . 4 𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
9 simpl 486 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
109a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝑦))
11 simpr 488 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
1211a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐵))
13 pm3.2an3 1337 . . . . . . 7 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1410, 12, 13syl6c 70 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
15 en3lp 9074 . . . . . 6 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
16 con3 156 . . . . . 6 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
1714, 15, 16syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝐵𝑥))
18 eleq2 2904 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
1918biimprcd 253 . . . . . . . 8 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2012, 19syl6 35 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵𝑦𝑥)))
21 pm3.2 473 . . . . . . 7 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2210, 20, 21syl10 79 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))))
23 en2lp 9066 . . . . . 6 ¬ (𝑥𝑦𝑦𝑥)
24 con3 156 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
2522, 23, 24syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝑥 = 𝐵))
26 simp3 1135 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
27 simp1 1133 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
28 trel 5165 . . . . . . . . . . 11 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
2928expd 419 . . . . . . . . . 10 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
3027, 12, 26, 29ee121 41131 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐴))
31 trel 5165 . . . . . . . . . 10 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
3231expd 419 . . . . . . . . 9 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
3327, 10, 30, 32ee122 41132 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐴))
34 ralcom 3345 . . . . . . . . . 10 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3534biimpi 219 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
36353ad2ant2 1131 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
37 rspsbc2 41160 . . . . . . . 8 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
3826, 33, 36, 37ee121 41131 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
39 equid 2020 . . . . . . . 8 𝑥 = 𝑥
40 sbceq1a 3769 . . . . . . . 8 (𝑥 = 𝑥 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
4139, 40ax-mp 5 . . . . . . 7 ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
4238, 41syl6ibr 255 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
43 sbcoreleleq 41161 . . . . . . 7 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4443biimpd 232 . . . . . 6 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4526, 42, 44sylsyld 61 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
46 3ornot23 41135 . . . . . 6 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
4746ex 416 . . . . 5 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
4817, 25, 45, 47ee222 41128 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
498, 48alrimi 2215 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
504, 49alrimi 2215 . 2 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
51 dftr2 5160 . 2 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
5250, 51sylibr 237 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3o 1083  w3a 1084  wal 1536   = wceq 1538  wcel 2115  wral 3133  [wsbc 3758  Tr wtr 5158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455  ax-reg 9053
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-fr 5501
This theorem is referenced by:  ordelordALT  41163  ordelordALTVD  41493
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