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Theorem intssunim 3851
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 3500 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
21ex 114 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
3 vex 2733 . . . 4 𝑦 ∈ V
43elint2 3836 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
5 eluni2 3798 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 4, 53imtr4g 204 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝐴𝑦 𝐴))
76ssrdv 3153 1 (∃𝑥 𝑥𝐴 𝐴 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1485  wcel 2141  wral 2448  wrex 2449  wss 3121   cuni 3794   cint 3829
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-int 3830
This theorem is referenced by:  intssuni2m  3853
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