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Theorem trintALT 44901
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 44901 is an alternate proof of trint 5277. trintALT 44901 is trintALTVD 44900 without virtual deductions and was automatically derived from trintALTVD 44900 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 482 . . . . 5 ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦)
21a1i 11 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧𝑦))
3 iidn3 44521 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑞𝐴)))
4 id 22 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥)
5 rspsbc 3879 . . . . . . . 8 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
63, 4, 5ee31 44772 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥)))
7 trsbc 44560 . . . . . . . 8 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
87biimpd 229 . . . . . . 7 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
93, 6, 8ee33 44541 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴 → Tr 𝑞)))
10 simpr 484 . . . . . . . . 9 ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴)
1110a1i 11 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑦 𝐴))
12 elintg 4954 . . . . . . . . 9 (𝑦 𝐴 → (𝑦 𝐴 ↔ ∀𝑞𝐴 𝑦𝑞))
1312ibi 267 . . . . . . . 8 (𝑦 𝐴 → ∀𝑞𝐴 𝑦𝑞)
1411, 13syl6 35 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞𝐴 𝑦𝑞))
15 rsp 3247 . . . . . . 7 (∀𝑞𝐴 𝑦𝑞 → (𝑞𝐴𝑦𝑞))
1614, 15syl6 35 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑦𝑞)))
17 trel 5268 . . . . . . 7 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
1817expd 415 . . . . . 6 (Tr 𝑞 → (𝑧𝑦 → (𝑦𝑞𝑧𝑞)))
199, 2, 16, 18ee323 44528 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑧𝑞)))
2019ralrimdv 3152 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞𝐴 𝑧𝑞))
21 elintg 4954 . . . . 5 (𝑧𝑦 → (𝑧 𝐴 ↔ ∀𝑞𝐴 𝑧𝑞))
2221biimprd 248 . . . 4 (𝑧𝑦 → (∀𝑞𝐴 𝑧𝑞𝑧 𝐴))
232, 20, 22syl6c 70 . . 3 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2423alrimivv 1928 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
25 dftr2 5261 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
2624, 25sylibr 234 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538  wcel 2108  wral 3061  [wsbc 3788   cint 4946  Tr wtr 5259
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-sbc 3789  df-ss 3968  df-uni 4908  df-int 4947  df-tr 5260
This theorem is referenced by: (None)
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