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Theorem tratrb 39248
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 39596. (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 1992 . . . 4 𝑥Tr 𝐴
2 nfra1 3079 . . . 4 𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
3 nfv 1992 . . . 4 𝑥 𝐵𝐴
41, 2, 3nf3an 1980 . . 3 𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
5 nfv 1992 . . . . 5 𝑦Tr 𝐴
6 nfra2 3084 . . . . 5 𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
7 nfv 1992 . . . . 5 𝑦 𝐵𝐴
85, 6, 7nf3an 1980 . . . 4 𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
9 simpl 474 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
109a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝑦))
11 simpr 479 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
1211a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐵))
13 pm3.2an3 1425 . . . . . . 7 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1410, 12, 13syl6c 70 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
15 en3lp 8682 . . . . . 6 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
16 con3 149 . . . . . 6 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
1714, 15, 16syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝐵𝑥))
18 eleq2 2828 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
1918biimprcd 240 . . . . . . . 8 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2012, 19syl6 35 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵𝑦𝑥)))
21 pm3.2 462 . . . . . . 7 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2210, 20, 21syl10 79 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))))
23 en2lp 8675 . . . . . 6 ¬ (𝑥𝑦𝑦𝑥)
24 con3 149 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
2522, 23, 24syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝑥 = 𝐵))
26 simp3 1133 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
27 simp1 1131 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
28 trel 4911 . . . . . . . . . . 11 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
2928expd 451 . . . . . . . . . 10 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
3027, 12, 26, 29ee121 39213 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐴))
31 trel 4911 . . . . . . . . . 10 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
3231expd 451 . . . . . . . . 9 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
3327, 10, 30, 32ee122 39214 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐴))
34 ralcom2 3242 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
35343ad2ant2 1129 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
36 rspsbc2 39246 . . . . . . . 8 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
3726, 33, 35, 36ee121 39213 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
38 equid 2094 . . . . . . . 8 𝑥 = 𝑥
39 sbceq1a 3587 . . . . . . . 8 (𝑥 = 𝑥 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
4038, 39ax-mp 5 . . . . . . 7 ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
4137, 40syl6ibr 242 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
42 sbcoreleleq 39247 . . . . . . 7 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4342biimpd 219 . . . . . 6 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4426, 41, 43sylsyld 61 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
45 3ornot23 39217 . . . . . 6 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
4645ex 449 . . . . 5 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
4717, 25, 44, 46ee222 39210 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
488, 47alrimi 2229 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
494, 48alrimi 2229 . 2 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
50 dftr2 4906 . 2 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
5149, 50sylibr 224 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1071  w3a 1072  wal 1630   = wceq 1632  wcel 2139  wral 3050  [wsbc 3576  Tr wtr 4904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114  ax-reg 8662
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-fr 5225
This theorem is referenced by:  ordelordALT  39249  ordelordALTVD  39602
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