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Theorem xpsspwOLD 4988
 Description: A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpsspwOLD

Proof of Theorem xpsspwOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4984 . 2
2 opelxp 4909 . . 3
3 snssi 3943 . . . . . . . 8
4 ssun3 3513 . . . . . . . 8
53, 4syl 16 . . . . . . 7
6 snex 4406 . . . . . . . 8
76elpw 3806 . . . . . . 7
85, 7sylibr 205 . . . . . 6
98adantr 453 . . . . 5
10 df-pr 3822 . . . . . . 7
11 snssi 3943 . . . . . . . . . 10
12 ssun4 3514 . . . . . . . . . 10
1311, 12syl 16 . . . . . . . . 9
145, 13anim12i 551 . . . . . . . 8
15 unss 3522 . . . . . . . 8
1614, 15sylib 190 . . . . . . 7
1710, 16syl5eqss 3393 . . . . . 6
18 zfpair2 4405 . . . . . . 7
1918elpw 3806 . . . . . 6
2017, 19sylibr 205 . . . . 5
219, 20jca 520 . . . 4
22 prex 4407 . . . . . 6
2322elpw 3806 . . . . 5
24 vex 2960 . . . . . . 7
25 vex 2960 . . . . . . 7
2624, 25dfop 3984 . . . . . 6
2726eleq1i 2500 . . . . 5
286, 18prss 3953 . . . . 5
2923, 27, 283bitr4ri 271 . . . 4
3021, 29sylib 190 . . 3
312, 30sylbi 189 . 2
321, 31relssi 4968 1
 Colors of variables: wff set class Syntax hints:   wa 360   wcel 1726   cun 3319   wss 3321  cpw 3800  csn 3815  cpr 3816  cop 3818   cxp 4877 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-opab 4268  df-xp 4885  df-rel 4886
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