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Theorem trintssm 3870
 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 𝑥𝐴 ∧ 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 2560 . . . 4 𝑦 ∈ V
21elint2 3622 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2m 3309 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 108 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 3861 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 1330 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 2432 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 389 . . 3 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 141 . 2 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 2951 1 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ∩ cint 3615  Tr wtr 3854 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616  df-tr 3855 This theorem is referenced by: (None)
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