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Theorem intssunim 3763
Description: The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssunim (∃𝑥 𝑥𝐴 𝐴 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem intssunim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.2m 3419 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
21ex 114 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
3 vex 2663 . . . 4 𝑦 ∈ V
43elint2 3748 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
5 eluni2 3710 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 4, 53imtr4g 204 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝐴𝑦 𝐴))
76ssrdv 3073 1 (∃𝑥 𝑥𝐴 𝐴 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1453  wcel 1465  wral 2393  wrex 2394  wss 3041   cuni 3706   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-int 3742
This theorem is referenced by:  intssuni2m  3765
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