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Theorem intssunim 3628
Description: The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssunim (x x A 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 3303 . . . 4 ((x x A x A y x) → x A y x)
21ex 108 . . 3 (x x A → (x A y xx A y x))
3 vex 2554 . . . 4 y V
43elint2 3613 . . 3 (y Ax A y x)
5 eluni2 3575 . . 3 (y Ax A y x)
62, 4, 53imtr4g 194 . 2 (x x A → (y Ay A))
76ssrdv 2945 1 (x x A A A)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1378   wcel 1390  wral 2300  wrex 2301  wss 2911   cuni 3571   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-int 3607
This theorem is referenced by:  intssuni2m  3630
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