![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7225 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
2 | 1 | anidms 562 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 Vcvv 3414 × cxp 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-opab 4938 df-xp 5352 df-rel 5353 |
This theorem is referenced by: resiexg 7369 erex 8038 hartogslem2 8724 harwdom 8771 dfac8b 9174 ac10ct 9177 canthwe 9795 brcic 16817 ciclcl 16821 cicrcl 16822 cicer 16825 ssclem 16838 estrccofval 17128 ipolerval 17516 mat0op 20599 matecl 20605 matlmod 20609 mattposvs 20636 ustval 22383 isust 22384 restutopopn 22419 ressuss 22444 ispsmet 22486 ismet 22505 isxmet 22506 bj-diagval 33614 fin2so 33934 rtrclexlem 38759 isclintop 42704 isassintop 42707 dfrngc2 42833 rngccofvalALTV 42848 dfringc2 42879 rngcresringcat 42891 ringccofvalALTV 42911 |
Copyright terms: Public domain | W3C validator |