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Theorem trintALT 40634
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 40634 is an alternate proof of trint 5040. trintALT 40634 is trintALTVD 40633 without virtual deductions and was automatically derived from trintALTVD 40633 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintALT (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintALT
Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 475 . . . . 5 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
21a1i 11 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦))
3 iidn3 40254 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑞𝐴)))
4 id 22 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥)
5 rspsbc 3759 . . . . . . . 8 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
63, 4, 5ee31 40505 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥)))
7 trsbc 40293 . . . . . . . 8 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
87biimpd 221 . . . . . . 7 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
93, 6, 8ee33 40274 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴 → Tr 𝑞)))
10 simpr 477 . . . . . . . . 9 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
1110a1i 11 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴))
12 elintg 4751 . . . . . . . . 9 (𝑦 𝐴 → (𝑦 𝐴 ↔ ∀𝑞𝐴 𝑦𝑞))
1312ibi 259 . . . . . . . 8 (𝑦 𝐴 → ∀𝑞𝐴 𝑦𝑞)
1411, 13syl6 35 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞𝐴 𝑦𝑞))
15 rsp 3149 . . . . . . 7 (∀𝑞𝐴 𝑦𝑞 → (𝑞𝐴𝑦𝑞))
1614, 15syl6 35 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑦𝑞)))
17 trel 5031 . . . . . . 7 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
1817expd 408 . . . . . 6 (Tr 𝑞 → (𝑧𝑦 → (𝑦𝑞𝑧𝑞)))
199, 2, 16, 18ee323 40261 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑧𝑞)))
2019ralrimdv 3132 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞𝐴 𝑧𝑞))
21 elintg 4751 . . . . 5 (𝑧𝑦 → (𝑧 𝐴 ↔ ∀𝑞𝐴 𝑧𝑞))
2221biimprd 240 . . . 4 (𝑧𝑦 → (∀𝑞𝐴 𝑧𝑞𝑧 𝐴))
232, 20, 22syl6c 70 . . 3 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2423alrimivv 1887 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
25 dftr2 5026 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2624, 25sylibr 226 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wal 1505  wcel 2050  wral 3082  [wsbc 3675   cint 4743  Tr wtr 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-v 3411  df-sbc 3676  df-in 3830  df-ss 3837  df-uni 4707  df-int 4744  df-tr 5025
This theorem is referenced by: (None)
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