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

Theorem truniALTVD 41218
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 40881 is truniALTVD 41218 without virtual deductions and was automatically derived from truniALTVD 41218.
 1:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 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(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ) 6:: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ) 7:6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   ) 8:6: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   ) 9:1,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   ) 10:8,9: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   ) 11:3,7,10: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   ) 12:11,8: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   ) 13:12: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 14:13: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 15:14: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 16:5,15: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   ) 17:16: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 18:17: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥    ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   ) 19:18: ⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∪ 𝐴   ) qed:19: ⊢ (∀𝑥 ∈ 𝐴Tr 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 40953 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   (𝑧𝑦𝑦 𝐴)   )
2 simpr 487 . . . . . . . 8 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
31, 2e2 40971 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑦 𝐴   )
4 eluni 4844 . . . . . . . 8 (𝑦 𝐴 ↔ ∃𝑞(𝑦𝑞𝑞𝐴))
54biimpi 218 . . . . . . 7 (𝑦 𝐴 → ∃𝑞(𝑦𝑞𝑞𝐴))
63, 5e2 40971 . . . . . 6 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞(𝑦𝑞𝑞𝐴)   )
7 simpl 485 . . . . . . . . . . . 12 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
81, 7e2 40971 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧𝑦   )
9 idn3 40955 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   (𝑦𝑞𝑞𝐴)   )
10 simpl 485 . . . . . . . . . . . 12 ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)
119, 10e3 41077 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑦𝑞   )
12 simpr 487 . . . . . . . . . . . . 13 ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)
139, 12e3 41077 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑞𝐴   )
14 idn1 40914 . . . . . . . . . . . . 13 (   𝑥𝐴 Tr 𝑥   ▶   𝑥𝐴 Tr 𝑥   )
15 rspsbc 3865 . . . . . . . . . . . . . 14 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
1615com12 32 . . . . . . . . . . . . 13 (∀𝑥𝐴 Tr 𝑥 → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥))
1714, 13, 16e13 41088 . . . . . . . . . . . 12 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
18 trsbc 40880 . . . . . . . . . . . . 13 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
1918biimpd 231 . . . . . . . . . . . 12 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
2013, 17, 19e33 41074 . . . . . . . . . . 11 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   Tr 𝑞   )
21 trel 5182 . . . . . . . . . . . 12 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
2221expdcom 417 . . . . . . . . . . 11 (𝑧𝑦 → (𝑦𝑞 → (Tr 𝑞𝑧𝑞)))
238, 11, 20, 22e233 41105 . . . . . . . . . 10 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑧𝑞   )
24 elunii 4846 . . . . . . . . . . 11 ((𝑧𝑞𝑞𝐴) → 𝑧 𝐴)
2524ex 415 . . . . . . . . . 10 (𝑧𝑞 → (𝑞𝐴𝑧 𝐴))
2623, 13, 25e33 41074 . . . . . . . . 9 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ,   (𝑦𝑞𝑞𝐴)   ▶   𝑧 𝐴   )
2726in3 40949 . . . . . . . 8 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
2827gen21 40959 . . . . . . 7 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)   )
29 19.23v 1942 . . . . . . . 8 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) ↔ (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
3029biimpi 218 . . . . . . 7 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) → (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
3128, 30e2 40971 . . . . . 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 41011 . . . . 5 (   𝑥𝐴 Tr 𝑥   ,   (𝑧𝑦𝑦 𝐴)   ▶   𝑧 𝐴   )
3433in2 40945 . . . 4 (   𝑥𝐴 Tr 𝑥   ▶   ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
3534gen12 40958 . . 3 (   𝑥𝐴 Tr 𝑥   ▶   𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴)   )
36 dftr2 5177 . . . 4 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
3736biimpri 230 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴) → Tr 𝐴)
3835, 37e1a 40967 . 2 (   𝑥𝐴 Tr 𝑥   ▶   Tr 𝐴   )
3938in1 40911 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779   ∈ wcel 2113  ∀wral 3141  [wsbc 3775  ∪ cuni 4841  Tr wtr 5175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-v 3499  df-sbc 3776  df-in 3946  df-ss 3955  df-uni 4842  df-tr 5176  df-vd1 40910  df-vd2 40918  df-vd3 40930 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator