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Theorem trintss 4697
 Description: If 𝐴 is transitive and non-null, then ∩ 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 3-Mar-2011.)
Assertion
Ref Expression
trintss ((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)

Proof of Theorem trintss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . 4 𝑦 ∈ V
21elint2 4417 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
3 r19.2z 4012 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
43ex 449 . . . 4 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
5 trel 4687 . . . . . 6 (Tr 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
65expcomd 453 . . . . 5 (Tr 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
76rexlimdv 3012 . . . 4 (Tr 𝐴 → (∃𝑥𝐴 𝑦𝑥𝑦𝐴))
84, 7sylan9 687 . . 3 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (∀𝑥𝐴 𝑦𝑥𝑦𝐴))
92, 8syl5bi 231 . 2 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (𝑦 𝐴𝑦𝐴))
109ssrdv 3574 1 ((𝐴 ≠ ∅ ∧ Tr 𝐴) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Tr wtr 4680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-uni 4373  df-int 4411  df-tr 4681 This theorem is referenced by: (None)
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