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Mirrors > Home > NFE Home > Th. List > elopab | GIF version |
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
elopab | ⊢ (A ∈ {〈x, y〉 ∣ φ} ↔ ∃x∃y(A = 〈x, y〉 ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2867 | . 2 ⊢ (A ∈ {〈x, y〉 ∣ φ} → A ∈ V) | |
2 | vex 2862 | . . . . . 6 ⊢ x ∈ V | |
3 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
4 | 2, 3 | opex 4588 | . . . . 5 ⊢ 〈x, y〉 ∈ V |
5 | eleq1 2413 | . . . . 5 ⊢ (A = 〈x, y〉 → (A ∈ V ↔ 〈x, y〉 ∈ V)) | |
6 | 4, 5 | mpbiri 224 | . . . 4 ⊢ (A = 〈x, y〉 → A ∈ V) |
7 | 6 | adantr 451 | . . 3 ⊢ ((A = 〈x, y〉 ∧ φ) → A ∈ V) |
8 | 7 | exlimivv 1635 | . 2 ⊢ (∃x∃y(A = 〈x, y〉 ∧ φ) → A ∈ V) |
9 | eqeq1 2359 | . . . . 5 ⊢ (z = A → (z = 〈x, y〉 ↔ A = 〈x, y〉)) | |
10 | 9 | anbi1d 685 | . . . 4 ⊢ (z = A → ((z = 〈x, y〉 ∧ φ) ↔ (A = 〈x, y〉 ∧ φ))) |
11 | 10 | 2exbidv 1628 | . . 3 ⊢ (z = A → (∃x∃y(z = 〈x, y〉 ∧ φ) ↔ ∃x∃y(A = 〈x, y〉 ∧ φ))) |
12 | df-opab 4623 | . . 3 ⊢ {〈x, y〉 ∣ φ} = {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} | |
13 | 11, 12 | elab2g 2987 | . 2 ⊢ (A ∈ V → (A ∈ {〈x, y〉 ∣ φ} ↔ ∃x∃y(A = 〈x, y〉 ∧ φ))) |
14 | 1, 8, 13 | pm5.21nii 342 | 1 ⊢ (A ∈ {〈x, y〉 ∣ φ} ↔ ∃x∃y(A = 〈x, y〉 ∧ φ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 〈cop 4561 {copab 4622 |
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 df-opab 4623 |
This theorem is referenced by: opelopabt 4699 opelopabga 4700 opabn0 4716 el1st 4729 setconslem4 4734 elswap 4740 elxp 4801 elcnv 4889 composeex 5820 |
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