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Mirrors > Home > NFE Home > Th. List > opeqexb | GIF version |
Description: A class is a set iff it is equal to an ordered pair. (Contributed by Scott Fenton, 19-Apr-2021.) |
Ref | Expression |
---|---|
opeqexb | ⊢ (A ∈ V ↔ ∃x∃y A = 〈x, y〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opexb 4603 | . 2 ⊢ (〈 Proj1 A, Proj2 A〉 ∈ V ↔ ( Proj1 A ∈ V ∧ Proj2 A ∈ V)) | |
2 | opeq 4619 | . . 3 ⊢ A = 〈 Proj1 A, Proj2 A〉 | |
3 | 2 | eleq1i 2416 | . 2 ⊢ (A ∈ V ↔ 〈 Proj1 A, Proj2 A〉 ∈ V) |
4 | eeanv 1913 | . . 3 ⊢ (∃x∃y(x = Proj1 A ∧ y = Proj2 A) ↔ (∃x x = Proj1 A ∧ ∃y y = Proj2 A)) | |
5 | 2 | eqeq1i 2360 | . . . . 5 ⊢ (A = 〈x, y〉 ↔ 〈 Proj1 A, Proj2 A〉 = 〈x, y〉) |
6 | eqcom 2355 | . . . . 5 ⊢ (〈 Proj1 A, Proj2 A〉 = 〈x, y〉 ↔ 〈x, y〉 = 〈 Proj1 A, Proj2 A〉) | |
7 | opth 4602 | . . . . 5 ⊢ (〈x, y〉 = 〈 Proj1 A, Proj2 A〉 ↔ (x = Proj1 A ∧ y = Proj2 A)) | |
8 | 5, 6, 7 | 3bitri 262 | . . . 4 ⊢ (A = 〈x, y〉 ↔ (x = Proj1 A ∧ y = Proj2 A)) |
9 | 8 | 2exbii 1583 | . . 3 ⊢ (∃x∃y A = 〈x, y〉 ↔ ∃x∃y(x = Proj1 A ∧ y = Proj2 A)) |
10 | isset 2863 | . . . 4 ⊢ ( Proj1 A ∈ V ↔ ∃x x = Proj1 A) | |
11 | isset 2863 | . . . 4 ⊢ ( Proj2 A ∈ V ↔ ∃y y = Proj2 A) | |
12 | 10, 11 | anbi12i 678 | . . 3 ⊢ (( Proj1 A ∈ V ∧ Proj2 A ∈ V) ↔ (∃x x = Proj1 A ∧ ∃y y = Proj2 A)) |
13 | 4, 9, 12 | 3bitr4i 268 | . 2 ⊢ (∃x∃y A = 〈x, y〉 ↔ ( Proj1 A ∈ V ∧ Proj2 A ∈ V)) |
14 | 1, 3, 13 | 3bitr4i 268 | 1 ⊢ (A ∈ V ↔ ∃x∃y A = 〈x, y〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 〈cop 4561 Proj1 cproj1 4563 Proj2 cproj2 4564 |
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 |
This theorem is referenced by: opeqex 4621 eliunxp 4821 dmsnn0 5064 |
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