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Theorem truniALT 44453
Description: The union of a class of transitive sets is transitive. Alternate proof of truni 5302. truniALT 44453 is truniALTVD 44790 without virtual deductions and was automatically derived from truniALTVD 44790 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
truniALT (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem truniALT
Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . 6 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
21a1i 11 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴))
3 eluni 4934 . . . . 5 (𝑦 𝐴 ↔ ∃𝑞(𝑦𝑞𝑞𝐴))
42, 3imbitrdi 251 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∃𝑞(𝑦𝑞𝑞𝐴)))
5 simpl 482 . . . . . . . . 9 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
65a1i 11 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦))
7 simpl 482 . . . . . . . . 9 ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)
872a1i 12 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)))
9 simpr 484 . . . . . . . . . 10 ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)
1092a1i 12 . . . . . . . . 9 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)))
11 rspsbc 3895 . . . . . . . . . . 11 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
1211com12 32 . . . . . . . . . 10 (∀𝑥𝐴 Tr 𝑥 → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥))
1310, 12syl6d 75 . . . . . . . . 9 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → [𝑞 / 𝑥]Tr 𝑥)))
14 trsbc 44452 . . . . . . . . . 10 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
1514biimpd 229 . . . . . . . . 9 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
1610, 13, 15ee33 44433 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → Tr 𝑞)))
17 trel 5295 . . . . . . . . 9 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
1817expdcom 414 . . . . . . . 8 (𝑧𝑦 → (𝑦𝑞 → (Tr 𝑞𝑧𝑞)))
196, 8, 16, 18ee233 44431 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑧𝑞)))
20 elunii 4936 . . . . . . . 8 ((𝑧𝑞𝑞𝐴) → 𝑧 𝐴)
2120ex 412 . . . . . . 7 (𝑧𝑞 → (𝑞𝐴𝑧 𝐴))
2219, 10, 21ee33 44433 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
2322alrimdv 1928 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
24 19.23v 1941 . . . . 5 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) ↔ (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
2523, 24imbitrdi 251 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
264, 25mpdd 43 . . 3 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2726alrimivv 1927 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
28 dftr2 5288 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2927, 28sylibr 234 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wex 1777  wcel 2103  wral 3063  [wsbc 3798   cuni 4931  Tr wtr 5286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-v 3484  df-sbc 3799  df-ss 3987  df-uni 4932  df-tr 5287
This theorem is referenced by: (None)
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