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Theorem trintssOLD 4803
Description: Obsolete version of trintss 4802 as of 30-Oct-2021. (Contributed by Scott Fenton, 3-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintssOLD ((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)

Proof of Theorem trintssOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . . 4 𝑦 ∈ V
21elint2 4514 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2z 4093 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 449 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 4792 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 453 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 3059 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 690 . . 3 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 232 . 2 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 3642 1 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   cint 4507  Tr wtr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-uni 4469  df-int 4508  df-tr 4786
This theorem is referenced by: (None)
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