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

Theorem trintALTVD 41959
Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 41960. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 41960 is trintALTVD 41959 without virtual deductions and was automatically derived from trintALTVD 41959.
 1:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴Tr 𝑥   ) 2:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ) 3:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   ) 4:2: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   ) 5:4: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞   ) 6:5: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   ) 7:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   ) 8:7,6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   ) 9:7,1: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   ) 10:7,9: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   Tr 𝑞   ) 11:10,3,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   ) 12:11: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   ) 13:12: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   ) 14:13: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞   ) 15:3,14: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   ) 16:15: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   ) 17:16: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   ) 18:17: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∩ 𝐴   ) qed:18: ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintALTVD (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintALTVD
Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn2 41692 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
2 simpl 486 . . . . . . 7 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
31, 2e2 41710 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
4 idn3 41694 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑞𝐴   )
5 idn1 41653 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
6 rspsbc 3785 . . . . . . . . . . . 12 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
74, 5, 6e31 41830 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
8 trsbc 41619 . . . . . . . . . . . 12 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
98biimpd 232 . . . . . . . . . . 11 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
104, 7, 9e33 41813 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   Tr 𝑞   )
11 simpr 488 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
121, 11e2 41710 . . . . . . . . . . . . 13 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
13 elintg 4846 . . . . . . . . . . . . . 14 (𝑦 𝐴 → (𝑦 𝐴 ↔ ∀𝑞𝐴 𝑦𝑞))
1413ibi 270 . . . . . . . . . . . . 13 (𝑦 𝐴 → ∀𝑞𝐴 𝑦𝑞)
1512, 14e2 41710 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴 𝑦𝑞   )
16 rsp 3134 . . . . . . . . . . . 12 (∀𝑞𝐴 𝑦𝑞 → (𝑞𝐴𝑦𝑞))
1715, 16e2 41710 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑦𝑞)   )
18 pm2.27 42 . . . . . . . . . . 11 (𝑞𝐴 → ((𝑞𝐴𝑦𝑞) → 𝑦𝑞))
194, 17, 18e32 41837 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑦𝑞   )
20 trel 5145 . . . . . . . . . . 11 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
2120expd 419 . . . . . . . . . 10 (Tr 𝑞 → (𝑧𝑦 → (𝑦𝑞𝑧𝑞)))
2210, 3, 19, 21e323 41845 . . . . . . . . 9 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑧𝑞   )
2322in3 41688 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑧𝑞)   )
2423gen21 41698 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑞𝐴𝑧𝑞)   )
25 df-ral 3075 . . . . . . . 8 (∀𝑞𝐴 𝑧𝑞 ↔ ∀𝑞(𝑞𝐴𝑧𝑞))
2625biimpri 231 . . . . . . 7 (∀𝑞(𝑞𝐴𝑧𝑞) → ∀𝑞𝐴 𝑧𝑞)
2724, 26e2 41710 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴 𝑧𝑞   )
28 elintg 4846 . . . . . . 7 (𝑧𝑦 → (𝑧 𝐴 ↔ ∀𝑞𝐴 𝑧𝑞))
2928biimprd 251 . . . . . 6 (𝑧𝑦 → (∀𝑞𝐴 𝑧𝑞𝑧 𝐴))
303, 27, 29e22 41750 . . . . 5 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
3130in2 41684 . . . 4 (   𝑥𝐴 Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
3231gen12 41697 . . 3 (   𝑥𝐴 Tr 𝑥   ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
33 dftr2 5140 . . . 4 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
3433biimpri 231 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴) → Tr 𝐴)
3532, 34e1a 41706 . 2 (   𝑥𝐴 Tr 𝑥   ▶   Tr 𝐴   )
3635in1 41650 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   ∈ wcel 2111  ∀wral 3070  [wsbc 3696  ∩ cint 4838  Tr wtr 5138 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-sbc 3697  df-in 3865  df-ss 3875  df-uni 4799  df-int 4839  df-tr 5139  df-vd1 41649  df-vd2 41657  df-vd3 41669 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator