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Theorem trintssm 3974
Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintssm
StepHypRef Expression
1 intss1 3725 . . . 4 (𝑥𝐴 𝐴𝑥)
2 trss 3967 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
32com12 30 . . . 4 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
4 sstr2 3046 . . . 4 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
51, 3, 4sylsyld 58 . . 3 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
65exlimiv 1541 . 2 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76impcom 124 1 ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1433  wcel 1445  wss 3013   cint 3710  Tr wtr 3958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-int 3711  df-tr 3959
This theorem is referenced by: (None)
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