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Theorem intssunim 3664
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 3336 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝑦𝑥) → ∃𝑥𝐴 𝑦𝑥)
21ex 112 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝑦𝑥 → ∃𝑥𝐴 𝑦𝑥))
3 vex 2577 . . . 4 𝑦 ∈ V
43elint2 3649 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
5 eluni2 3611 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
62, 4, 53imtr4g 198 . 2 (∃𝑥 𝑥𝐴 → (𝑦 𝐴𝑦 𝐴))
76ssrdv 2978 1 (∃𝑥 𝑥𝐴 𝐴 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1397  wcel 1409  wral 2323  wrex 2324  wss 2944   cuni 3607   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-uni 3608  df-int 3643
This theorem is referenced by:  intssuni2m  3666
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