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Theorem truniALTVD 39637
Description: The union of a class of transitive sets is transitive. 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. truniALT 39277 is truniALTVD 39637 without virtual deductions and was automatically derived from truniALTVD 39637.
1:: (   𝑥𝐴Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
2:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
3:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑧𝑦   )
4:2: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑦 𝐴   )
5:4: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑞(𝑦𝑞𝑞𝐴)   )
6:: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   (𝑦𝑞𝑞𝐴)   )
7:6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑦𝑞   )
8:6: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑞𝐴   )
9:1,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:8,9: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   Tr 𝑞   )
11:3,7,10: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑧𝑞   )
12:11,8: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴), (𝑦𝑞𝑞𝐴)   ▶   𝑧 𝐴   )
13:12: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
14:13: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
15:14: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
16:5,15: (   𝑥𝐴Tr 𝑥   ,   (𝑧𝑦 𝑦 𝐴)   ▶   𝑧 𝐴   )
17:16: (   𝑥𝐴Tr 𝑥   ▶   ((𝑧𝑦 𝑦 𝐴) → 𝑧 𝐴)   )
18:17: (   𝑥𝐴Tr 𝑥    ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
19:18: (   𝑥𝐴Tr 𝑥   ▶   Tr 𝐴   )
qed:19: (∀𝑥𝐴Tr 𝑥 → Tr 𝐴)
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
truniALTVD (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem truniALTVD
Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn2 39364 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
2 simpr 471 . . . . . . . 8 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
31, 2e2 39382 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
4 eluni 4578 . . . . . . . 8 (𝑦 𝐴 ↔ ∃𝑞(𝑦𝑞𝑞𝐴))
54biimpi 206 . . . . . . 7 (𝑦 𝐴 → ∃𝑞(𝑦𝑞𝑞𝐴))
63, 5e2 39382 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑦𝑞𝑞𝐴)   )
7 simpl 468 . . . . . . . . . . . 12 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
81, 7e2 39382 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
9 idn3 39366 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   (𝑦𝑞𝑞𝐴)   )
10 simpl 468 . . . . . . . . . . . 12 ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)
119, 10e3 39490 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑦𝑞   )
12 simpr 471 . . . . . . . . . . . . 13 ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)
139, 12e3 39490 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑞𝐴   )
14 idn1 39316 . . . . . . . . . . . . 13 (   𝑥𝐴 Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
15 rspsbc 3668 . . . . . . . . . . . . . 14 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
1615com12 32 . . . . . . . . . . . . 13 (∀𝑥𝐴 Tr 𝑥 → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥))
1714, 13, 16e13 39501 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
18 trsbc 39276 . . . . . . . . . . . . 13 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
1918biimpd 219 . . . . . . . . . . . 12 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
2013, 17, 19e33 39487 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   Tr 𝑞   )
21 trel 4894 . . . . . . . . . . . 12 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
2221expdcom 399 . . . . . . . . . . 11 (𝑧𝑦 → (𝑦𝑞 → (Tr 𝑞𝑧𝑞)))
238, 11, 20, 22e233 39518 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑧𝑞   )
24 elunii 4580 . . . . . . . . . . 11 ((𝑧𝑞𝑞𝐴) → 𝑧 𝐴)
2524ex 397 . . . . . . . . . 10 (𝑧𝑞 → (𝑞𝐴𝑧 𝐴))
2623, 13, 25e33 39487 . . . . . . . . 9 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑧 𝐴   )
2726in3 39360 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
2827gen21 39370 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
29 19.23v 2023 . . . . . . . 8 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) ↔ (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
3029biimpi 206 . . . . . . 7 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) → (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
3128, 30e2 39382 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
32 pm2.27 42 . . . . . 6 (∃𝑞(𝑦𝑞𝑞𝐴) → ((∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴) → 𝑧 𝐴))
336, 31, 32e22 39422 . . . . 5 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
3433in2 39356 . . . 4 (   𝑥𝐴 Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
3534gen12 39369 . . 3 (   𝑥𝐴 Tr 𝑥   ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
36 dftr2 4889 . . . 4 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
3736biimpri 218 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴) → Tr 𝐴)
3835, 37e1a 39378 . 2 (   𝑥𝐴 Tr 𝑥   ▶   Tr 𝐴   )
3938in1 39313 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wex 1852  wcel 2145  wral 3061  [wsbc 3588   cuni 4575  Tr wtr 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-sbc 3589  df-in 3731  df-ss 3738  df-uni 4576  df-tr 4888  df-vd1 39312  df-vd2 39320  df-vd3 39332
This theorem is referenced by: (None)
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