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Theorem trintssmOLD 3953
 Description: Obsolete version of trintssm 3952 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintssmOLD ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → 𝐴𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintssmOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . . 4 𝑦 ∈ V
21elint2 3695 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2m 3369 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 113 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 3943 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 1375 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 2488 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 401 . . 3 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 150 . 2 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 3031 1 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  ∃wex 1426   ∈ wcel 1438  ∀wral 2359  ∃wrex 2360   ⊆ wss 2999  ∩ cint 3688  Tr wtr 3936 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070 This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-int 3689  df-tr 3937 This theorem is referenced by: (None)
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