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Theorem trintALT 44878
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 44878 is an alternate proof of trint 5282. trintALT 44878 is trintALTVD 44877 without virtual deductions and was automatically derived from trintALTVD 44877 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 44498 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑞𝐴)))
4 id 22 . . . . . . . 8 (∀𝑥𝐴 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥)
5 rspsbc 3887 . . . . . . . 8 (𝑞𝐴 → (∀𝑥𝐴 Tr 𝑥[𝑞 / 𝑥]Tr 𝑥))
63, 4, 5ee31 44749 . . . . . . 7 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴[𝑞 / 𝑥]Tr 𝑥)))
7 trsbc 44537 . . . . . . . 8 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
87biimpd 229 . . . . . . 7 (𝑞𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
93, 6, 8ee33 44518 . . . . . 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 4958 . . . . . . . . 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 3244 . . . . . . 7 (∀𝑞𝐴 𝑦𝑞 → (𝑞𝐴𝑦𝑞))
1614, 15syl6 35 . . . . . 6 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑦𝑞)))
17 trel 5273 . . . . . . 7 (Tr 𝑞 → ((𝑧𝑦𝑦𝑞) → 𝑧𝑞))
1817expd 415 . . . . . 6 (Tr 𝑞 → (𝑧𝑦 → (𝑦𝑞𝑧𝑞)))
199, 2, 16, 18ee323 44505 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → (𝑞𝐴𝑧𝑞)))
2019ralrimdv 3149 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ((𝑧𝑦𝑦 𝐴) → ∀𝑞𝐴 𝑧𝑞))
21 elintg 4958 . . . . 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 1925 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑧𝑦((𝑧𝑦𝑦 𝐴) → 𝑧 𝐴))
25 dftr2 5266 . 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 1534  wcel 2105  wral 3058  [wsbc 3790   cint 4950  Tr wtr 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-v 3479  df-sbc 3791  df-ss 3979  df-uni 4912  df-int 4951  df-tr 5265
This theorem is referenced by: (None)
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