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Theorem truniALT 42915
Description: The union of a class of transitive sets is transitive. Alternate proof of truni 5242. truniALT 42915 is truniALTVD 43252 without virtual deductions and was automatically derived from truniALTVD 43252 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 486 . . . . . 6 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
21a1i 11 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴))
3 eluni 4872 . . . . 5 (𝑦 𝐴 ↔ ∃𝑞(𝑦𝑞𝑞𝐴))
42, 3syl6ib 251 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∃𝑞(𝑦𝑞𝑞𝐴)))
5 simpl 484 . . . . . . . . 9 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
65a1i 11 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦))
7 simpl 484 . . . . . . . . 9 ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)
872a1i 12 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑦𝑞)))
9 simpr 486 . . . . . . . . . 10 ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)
1092a1i 12 . . . . . . . . 9 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑞𝐴)))
11 rspsbc 3839 . . . . . . . . . . 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 42914 . . . . . . . . . 10 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
1514biimpd 228 . . . . . . . . 9 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
1610, 13, 15ee33 42895 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → Tr 𝑞)))
17 trel 5235 . . . . . . . . 9 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
1817expdcom 416 . . . . . . . 8 (𝑧𝑦 → (𝑦𝑞 → (Tr 𝑞𝑧𝑞)))
196, 8, 16, 18ee233 42893 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑧𝑞)))
20 elunii 4874 . . . . . . . 8 ((𝑧𝑞𝑞𝐴) → 𝑧 𝐴)
2120ex 414 . . . . . . 7 (𝑧𝑞 → (𝑞𝐴𝑧 𝐴))
2219, 10, 21ee33 42895 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
2322alrimdv 1933 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
24 19.23v 1946 . . . . 5 (∀𝑞((𝑦𝑞𝑞𝐴) → 𝑧 𝐴) ↔ (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴))
2523, 24syl6ib 251 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (∃𝑞(𝑦𝑞𝑞𝐴) → 𝑧 𝐴)))
264, 25mpdd 43 . . 3 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2726alrimivv 1932 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
28 dftr2 5228 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2927, 28sylibr 233 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wex 1782  wcel 2107  wral 3061  [wsbc 3743   cuni 4869  Tr wtr 5226
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-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 3062  df-v 3449  df-sbc 3744  df-in 3921  df-ss 3931  df-uni 4870  df-tr 5227
This theorem is referenced by: (None)
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