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Theorem trintssOLD 4768
 Description: Obsolete version of trintss 4767 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 3201 . . . 4 𝑦 ∈ V
21elint2 4480 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2z 4058 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 450 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 4757 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 454 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 3028 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 689 . . 3 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 232 . 2 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 3607 1 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  ∃wrex 2912   ⊆ wss 3572  ∅c0 3913  ∩ cint 4473  Tr wtr 4750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-nul 3914  df-uni 4435  df-int 4474  df-tr 4751 This theorem is referenced by: (None)
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