Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tratrb Structured version   Visualization version   GIF version

Theorem tratrb 44533
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 44858. (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 1911 . . . 4 𝑥Tr 𝐴
2 nfra1 3281 . . . 4 𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
3 nfv 1911 . . . 4 𝑥 𝐵𝐴
41, 2, 3nf3an 1898 . . 3 𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
5 nfv 1911 . . . . 5 𝑦Tr 𝐴
6 nfra2w 3296 . . . . 5 𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)
7 nfv 1911 . . . . 5 𝑦 𝐵𝐴
85, 6, 7nf3an 1898 . . . 4 𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)
9 simpl 482 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
109a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝑦))
11 simpr 484 . . . . . . . 8 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
1211a1i 11 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐵))
13 pm3.2an3 1339 . . . . . . 7 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1410, 12, 13syl6c 70 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
15 en3lp 9651 . . . . . 6 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
16 con3 153 . . . . . 6 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
1714, 15, 16syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝐵𝑥))
18 eleq2 2827 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
1918biimprcd 250 . . . . . . . 8 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2012, 19syl6 35 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵𝑦𝑥)))
21 pm3.2 469 . . . . . . 7 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2210, 20, 21syl10 79 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))))
23 en2lp 9643 . . . . . 6 ¬ (𝑥𝑦𝑦𝑥)
24 con3 153 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
2522, 23, 24syl6mpi 67 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → ¬ 𝑥 = 𝐵))
26 simp3 1137 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
27 simp1 1135 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
28 trel 5273 . . . . . . . . . . 11 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
2928expd 415 . . . . . . . . . 10 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
3027, 12, 26, 29ee121 44502 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑦𝐴))
31 trel 5273 . . . . . . . . . 10 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
3231expd 415 . . . . . . . . 9 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
3327, 10, 30, 32ee122 44503 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐴))
34 ralcom 3286 . . . . . . . . . 10 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3534biimpi 216 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
36353ad2ant2 1133 . . . . . . . 8 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
37 rspsbc2 44531 . . . . . . . 8 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
3826, 33, 36, 37ee121 44502 . . . . . . 7 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
39 equid 2008 . . . . . . . 8 𝑥 = 𝑥
40 sbceq1a 3801 . . . . . . . 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, 41imbitrrdi 252 . . . . . 6 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
43 sbcoreleleq 44532 . . . . . . 7 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4443biimpd 229 . . . . . 6 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
4526, 42, 44sylsyld 61 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
46 3ornot23 44506 . . . . . 6 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
4746ex 412 . . . . 5 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
4817, 25, 45, 47ee222 44499 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
498, 48alrimi 2210 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
504, 49alrimi 2210 . 2 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
51 dftr2 5266 . 2 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
5250, 51sylibr 234 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1085  w3a 1086  wal 1534   = wceq 1536  wcel 2105  wral 3058  [wsbc 3790  Tr wtr 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-reg 9629
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-fr 5640
This theorem is referenced by:  ordelordALT  44534  ordelordALTVD  44864
  Copyright terms: Public domain W3C validator