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Theorem intssunim 3793
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 3449 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
21ex 114 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
3 vex 2689 . . . 4 𝑦 ∈ V
43elint2 3778 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
5 eluni2 3740 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 4, 53imtr4g 204 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝐴𝑦 𝐴))
76ssrdv 3103 1 (∃𝑥 𝑥𝐴 𝐴 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1468  wcel 1480  wral 2416  wrex 2417  wss 3071   cuni 3736   cint 3771
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-int 3772
This theorem is referenced by:  intssuni2m  3795
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