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Theorem trintssmOLD 3899
Description: Obsolete version of trintssm 3898 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 2577 . . . 4 𝑦 ∈ V
21elint2 3650 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2m 3337 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 112 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 3889 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 1346 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 2449 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 395 . . 3 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 145 . 2 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 2979 1 ((∃𝑥 𝑥𝐴 ∧ Tr 𝐴) → 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  wcel 1409  wral 2323  wrex 2324  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-rex 2329  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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