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Theorem trint0m 3871
 Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trint0m ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trint0m
StepHypRef Expression
1 intss1 3630 . . . 4 (𝑥𝐴 𝐴𝑥)
2 trss 3863 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
32com12 27 . . . 4 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
4 sstr2 2952 . . . 4 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
51, 3, 4sylsyld 52 . . 3 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
65exlimiv 1489 . 2 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76impcom 116 1 ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393   ⊆ 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-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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