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Theorem trintssm 3898
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 3658 . . . 4 (𝑥𝐴 𝐴𝑥)
2 trss 3891 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
32com12 30 . . . 4 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
4 sstr2 2980 . . . 4 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
51, 3, 4sylsyld 56 . . 3 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
65exlimiv 1505 . 2 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76impcom 120 1 ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  wcel 1409  wss 2945   cint 3643  Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-int 3644  df-tr 3883
This theorem is referenced by: (None)
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