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Mirrors > Home > MPE Home > Th. List > sqxpexg | Structured version Visualization version GIF version |
Description: The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.) |
Ref | Expression |
---|---|
sqxpexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7731 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
2 | 1 | anidms 566 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 × cxp 5665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-opab 5202 df-xp 5673 df-rel 5674 |
This theorem is referenced by: resiexg 7899 erex 8724 hartogslem2 9535 harwdom 9583 dfac8b 10023 ac10ct 10026 canthwe 10643 cicer 17754 ssclem 17767 ipolerval 18489 dfrngc2 20516 dfringc2 20545 rngcresringcat 20557 mat0op 22245 matecl 22251 matlmod 22255 mattposvs 22281 ustval 24031 isust 24032 restutopopn 24067 ressuss 24091 ispsmet 24134 ismet 24153 isxmet 24154 satef 34898 satefvfmla0 34900 satefvfmla1 34907 fin2so 36969 rtrclexlem 42881 isclintop 47095 isassintop 47098 rngccofvalALTV 47158 ringccofvalALTV 47192 2arymaptf 47551 |
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