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Theorem trintALTVD 43641
Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 43642. 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 43642 is trintALTVD 43641 without virtual deductions and was automatically derived from trintALTVD 43641.
1:: (   𝑥𝐴Tr 𝑥   ▶   𝑥𝐴Tr 𝑥   )
2:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
3:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
4:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
5:4: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴𝑦𝑞   )
6:5: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑦𝑞)   )
7:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑞𝐴   )
8:7,6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑦𝑞   )
9:7,1: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:7,9: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   Tr 𝑞   )
11:10,3,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴), 𝑞𝐴   ▶   𝑧𝑞   )
12:11: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑧𝑞)   )
13:12: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑞𝐴𝑧𝑞)   )
14:13: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴𝑧𝑞   )
15:3,14: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
16:15: (   𝑥𝐴Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
17:16: (   𝑥𝐴Tr 𝑥   ▶   𝑧𝑦((𝑧 𝑦𝑦 𝐴) → 𝑧 𝐴)   )
18:17: (   𝑥𝐴Tr 𝑥   ▶   Tr 𝐴   )
qed:18: (∀𝑥𝐴Tr 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 43374 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
2 simpl 484 . . . . . . 7 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
31, 2e2 43392 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
4 idn3 43376 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑞𝐴   )
5 idn1 43335 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
6 rspsbc 3874 . . . . . . . . . . . 12 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
74, 5, 6e31 43512 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
8 trsbc 43301 . . . . . . . . . . . 12 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
98biimpd 228 . . . . . . . . . . 11 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
104, 7, 9e33 43495 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   Tr 𝑞   )
11 simpr 486 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
121, 11e2 43392 . . . . . . . . . . . . 13 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
13 elintg 4959 . . . . . . . . . . . . . 14 (𝑦 𝐴 → (𝑦 𝐴 ↔ ∀𝑞𝐴 𝑦𝑞))
1413ibi 267 . . . . . . . . . . . . 13 (𝑦 𝐴 → ∀𝑞𝐴 𝑦𝑞)
1512, 14e2 43392 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴 𝑦𝑞   )
16 rsp 3245 . . . . . . . . . . . 12 (∀𝑞𝐴 𝑦𝑞 → (𝑞𝐴𝑦𝑞))
1715, 16e2 43392 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑦𝑞)   )
18 pm2.27 42 . . . . . . . . . . 11 (𝑞𝐴 → ((𝑞𝐴𝑦𝑞) → 𝑦𝑞))
194, 17, 18e32 43519 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑦𝑞   )
20 trel 5275 . . . . . . . . . . 11 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
2120expd 417 . . . . . . . . . 10 (Tr 𝑞 → (𝑧𝑦 → (𝑦𝑞𝑧𝑞)))
2210, 3, 19, 21e323 43527 . . . . . . . . 9 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   𝑞𝐴   ▶   𝑧𝑞   )
2322in3 43370 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑞𝐴𝑧𝑞)   )
2423gen21 43380 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑞𝐴𝑧𝑞)   )
25 df-ral 3063 . . . . . . . 8 (∀𝑞𝐴 𝑧𝑞 ↔ ∀𝑞(𝑞𝐴𝑧𝑞))
2625biimpri 227 . . . . . . 7 (∀𝑞(𝑞𝐴𝑧𝑞) → ∀𝑞𝐴 𝑧𝑞)
2724, 26e2 43392 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞𝐴 𝑧𝑞   )
28 elintg 4959 . . . . . . 7 (𝑧𝑦 → (𝑧 𝐴 ↔ ∀𝑞𝐴 𝑧𝑞))
2928biimprd 247 . . . . . 6 (𝑧𝑦 → (∀𝑞𝐴 𝑧𝑞𝑧 𝐴))
303, 27, 29e22 43432 . . . . 5 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
3130in2 43366 . . . 4 (   𝑥𝐴 Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
3231gen12 43379 . . 3 (   𝑥𝐴 Tr 𝑥   ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
33 dftr2 5268 . . . 4 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
3433biimpri 227 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴) → Tr 𝐴)
3532, 34e1a 43388 . 2 (   𝑥𝐴 Tr 𝑥   ▶   Tr 𝐴   )
3635in1 43332 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  wral 3062  [wsbc 3778   cint 4951  Tr wtr 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-v 3477  df-sbc 3779  df-in 3956  df-ss 3966  df-uni 4910  df-int 4952  df-tr 5267  df-vd1 43331  df-vd2 43339  df-vd3 43351
This theorem is referenced by: (None)
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