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Mirrors > Home > NFE Home > Th. List > xpkex | GIF version |
Description: The Kuratowski cross product of two sets is a set. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
xpkex.1 | ⊢ A ∈ V |
xpkex.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
xpkex | ⊢ (A ×k B) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpkex.1 | . 2 ⊢ A ∈ V | |
2 | xpkex.2 | . 2 ⊢ B ∈ V | |
3 | xpkexg 4288 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ×k B) ∈ V) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ (A ×k B) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 ×k cxpk 4174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 df-cnvk 4186 |
This theorem is referenced by: sikexg 4296 imakexg 4299 ins2kexg 4305 ins3kexg 4306 imagekexg 4311 nnsucelrlem1 4424 ltfinex 4464 ssfin 4470 ncfinraiselem2 4480 ncfinlowerlem1 4482 tfinrelkex 4487 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 srelkex 4525 tfinnnlem1 4533 phiexg 4571 opexg 4587 proj1exg 4591 proj2exg 4592 setconslem5 4735 1stex 4739 swapex 4742 ssetex 4744 imaexg 4746 coexg 4749 siexg 4752 |
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