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Mirrors > Home > NFE Home > Th. List > opex | GIF version |
Description: An ordered pair of two sets is a set. (Contributed by SF, 5-Jan-2015.) |
Ref | Expression |
---|---|
opex.1 | ⊢ A ∈ V |
opex.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
opex | ⊢ 〈A, B〉 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex.1 | . 2 ⊢ A ∈ V | |
2 | opex.2 | . 2 ⊢ B ∈ V | |
3 | opexg 4587 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 ∈ V) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ 〈A, B〉 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 〈cop 4561 |
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-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 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-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 |
This theorem is referenced by: opabid 4695 elopab 4696 opabn0 4716 el1st 4729 setconslem6 4736 elswap 4740 raliunxp 4823 ssopr 4846 br1st 4858 br2nd 4859 brswap2 4860 intirr 5029 dmsnn0 5064 dmsnopg 5066 cnvsn 5073 opswap 5074 rnsnop 5075 dfcnv2 5100 df2nd2 5111 cnviin 5118 nfunv 5138 funsn 5147 fnasrn 5417 fsn 5432 fvsn 5445 brswap 5509 opfv1st 5514 opfv2nd 5515 oprabid 5550 dfoprab2 5558 rnoprab 5576 otelins2 5791 otelins3 5792 brimage 5793 oqelins4 5794 dmtxp 5802 otsnelsi3 5805 releqel 5807 releqmpt2 5809 composeex 5820 disjex 5823 addcfnex 5824 braddcfn 5826 funsex 5828 dmpprod 5840 fnfullfunlem1 5856 brfullfunop 5867 ranfnex 5871 transex 5910 refex 5911 antisymex 5912 connexex 5913 foundex 5914 extex 5915 symex 5916 qsexg 5982 xpassen 6057 enpw1lem1 6061 enmap2lem1 6063 enmap1lem1 6069 enprmaplem1 6076 ovmuc 6130 ovcelem1 6171 ceex 6174 tcfnex 6244 csucex 6259 addccan2nclem1 6263 nncdiv3lem1 6275 nncdiv3lem2 6276 nnc3n3p1 6278 spacvallem1 6281 nchoicelem10 6298 nchoicelem16 6304 frecxp 6314 dmfrec 6316 |
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