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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 7462 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
2 | 1 | anidms 567 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: resiexg 7608 erex 8302 hartogslem2 8995 harwdom 9042 dfac8b 9445 ac10ct 9448 canthwe 10061 ciclcl 17060 cicrcl 17061 cicer 17064 ssclem 17077 ipolerval 17754 mat0op 20956 matecl 20962 matlmod 20966 mattposvs 20992 ustval 22738 isust 22739 restutopopn 22774 ressuss 22799 ispsmet 22841 ismet 22860 isxmet 22861 satef 32560 satefvfmla0 32562 satefvfmla1 32569 fin2so 34760 rtrclexlem 39854 isclintop 44042 isassintop 44045 dfrngc2 44171 rngccofvalALTV 44186 dfringc2 44217 rngcresringcat 44229 ringccofvalALTV 44249 |
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