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Mirrors > Home > NFE Home > Th. List > ideq | GIF version |
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.) (Revised by set.mm contributors, 1-Jun-2008.) |
Ref | Expression |
---|---|
ideq.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
ideq | ⊢ (A I B ↔ A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ideq.1 | . 2 ⊢ B ∈ V | |
2 | ideqg 4868 | . 2 ⊢ (B ∈ V → (A I B ↔ A = B)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (A I B ↔ A = B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 Vcvv 2859 class class class wbr 4639 I cid 4763 |
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-opab 4623 df-br 4640 df-id 4767 |
This theorem is referenced by: dmi 4919 iss 5000 imai 5010 intasym 5028 intirr 5029 cnvi 5032 coi1 5094 dffun2 5119 fvi 5442 dfid4 5503 elfix 5787 composeex 5820 funsex 5828 pw1fnex 5852 fnfullfunlem1 5856 ranfnex 5871 antisymex 5912 connexex 5913 foundex 5914 extex 5915 symex 5916 ider 5943 idssen 6040 tcfnex 6244 fnfreclem1 6317 |
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