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Theorem uniixp 7077
 Description: The union of an infinite Cartesian product is included in a cross product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
uniixp
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem uniixp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ixpf 7076 . . . . 5
2 fssxp 5594 . . . . 5
31, 2syl 16 . . . 4
4 vex 2951 . . . . 5
54elpw 3797 . . . 4
63, 5sylibr 204 . . 3
76ssriv 3344 . 2
8 sspwuni 4168 . 2
97, 8mpbi 200 1
 Colors of variables: wff set class Syntax hints:   wcel 1725   wss 3312  cpw 3791  cuni 4007  ciun 4085   cxp 4868  wf 5442  cixp 7055 This theorem is referenced by:  ixpexg  7078 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ixp 7056
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