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Theorem trintssm 3860
 Description: If A is transitive and inhabited, then ∩ A is a subset of A. (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm ((x x A Tr A) → AA)
Distinct variable group:   x,A

Proof of Theorem trintssm
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . 4 y V
21elint2 3612 . . 3 (y Ax A y x)
3 r19.2m 3303 . . . . 5 ((x x A x A y x) → x A y x)
43ex 108 . . . 4 (x x A → (x A y xx A y x))
5 trel 3851 . . . . . 6 (Tr A → ((y x x A) → y A))
65expcomd 1327 . . . . 5 (Tr A → (x A → (y xy A)))
76rexlimdv 2426 . . . 4 (Tr A → (x A y xy A))
84, 7sylan9 389 . . 3 ((x x A Tr A) → (x A y xy A))
92, 8syl5bi 141 . 2 ((x x A Tr A) → (y Ay A))
109ssrdv 2945 1 ((x x A Tr A) → AA)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ∩ cint 3605  Tr wtr 3844 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3571  df-int 3606  df-tr 3845 This theorem is referenced by: (None)
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