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Theorem trintssm 4131
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 3873 . . . 4 (𝑥𝐴 𝐴𝑥)
2 trss 4124 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
32com12 30 . . . 4 (𝑥𝐴 → (Tr 𝐴𝑥𝐴))
4 sstr2 3176 . . . 4 ( 𝐴𝑥 → (𝑥𝐴 𝐴𝐴))
51, 3, 4sylsyld 58 . . 3 (𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
65exlimiv 1608 . 2 (∃𝑥 𝑥𝐴 → (Tr 𝐴 𝐴𝐴))
76impcom 125 1 ((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1502  wcel 2159  wss 3143   cint 3858  Tr wtr 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-v 2753  df-in 3149  df-ss 3156  df-uni 3824  df-int 3859  df-tr 4116
This theorem is referenced by: (None)
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