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Mirrors > Home > NFE Home > Th. List > brex | GIF version |
Description: Binary relationship implies sethood of both parts. (Contributed by SF, 7-Jan-2015.) |
Ref | Expression |
---|---|
brex | ⊢ (ARB → (A ∈ V ∧ B ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2867 | . 2 ⊢ (〈A, B〉 ∈ R → 〈A, B〉 ∈ V) | |
2 | df-br 4640 | . 2 ⊢ (ARB ↔ 〈A, B〉 ∈ R) | |
3 | opexb 4603 | . . 3 ⊢ (〈A, B〉 ∈ V ↔ (A ∈ V ∧ B ∈ V)) | |
4 | 3 | bicomi 193 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) ↔ 〈A, B〉 ∈ V) |
5 | 1, 2, 4 | 3imtr4i 257 | 1 ⊢ (ARB → (A ∈ V ∧ B ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2859 〈cop 4561 class class class wbr 4639 |
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-13 1712 ax-14 1714 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-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 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-pss 3261 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-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-br 4640 |
This theorem is referenced by: brreldmex 4690 brrelrnex 4691 opelopabsb 4697 elima 4754 brsi 4761 epelc 4765 vtoclr 4816 br1st 4858 br2nd 4859 brswap2 4860 ideqg 4868 ideqg2 4869 brco 4883 brcnv 4892 brres 4949 imasn 5018 elimasn 5019 fvprc 5325 opbr1st 5501 opbr2nd 5502 brswap 5509 fununiq 5517 trtxp 5781 elfix 5787 brfns 5833 qrpprod 5836 fnfullfunlem1 5856 clos1conn 5879 trd 5921 frd 5922 extd 5923 symd 5924 refd 5927 antid 5929 connexd 5931 frds 5935 bren 6030 enpw1 6062 enmap2 6068 enpw 6087 brltc 6114 elnc 6125 eqncg 6126 ncseqnc 6128 elce 6175 |
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