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Mirrors > Home > NFE Home > Th. List > opabn0 | GIF version |
Description: Non-empty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
opabn0 | ⊢ ({〈x, y〉 ∣ φ} ≠ ∅ ↔ ∃x∃yφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3559 | . 2 ⊢ ({〈x, y〉 ∣ φ} ≠ ∅ ↔ ∃z z ∈ {〈x, y〉 ∣ φ}) | |
2 | elopab 4696 | . . 3 ⊢ (z ∈ {〈x, y〉 ∣ φ} ↔ ∃x∃y(z = 〈x, y〉 ∧ φ)) | |
3 | 2 | exbii 1582 | . 2 ⊢ (∃z z ∈ {〈x, y〉 ∣ φ} ↔ ∃z∃x∃y(z = 〈x, y〉 ∧ φ)) |
4 | exrot3 1744 | . . 3 ⊢ (∃z∃x∃y(z = 〈x, y〉 ∧ φ) ↔ ∃x∃y∃z(z = 〈x, y〉 ∧ φ)) | |
5 | vex 2862 | . . . . . . 7 ⊢ x ∈ V | |
6 | vex 2862 | . . . . . . 7 ⊢ y ∈ V | |
7 | 5, 6 | opex 4588 | . . . . . 6 ⊢ 〈x, y〉 ∈ V |
8 | 7 | isseti 2865 | . . . . 5 ⊢ ∃z z = 〈x, y〉 |
9 | 19.41v 1901 | . . . . 5 ⊢ (∃z(z = 〈x, y〉 ∧ φ) ↔ (∃z z = 〈x, y〉 ∧ φ)) | |
10 | 8, 9 | mpbiran 884 | . . . 4 ⊢ (∃z(z = 〈x, y〉 ∧ φ) ↔ φ) |
11 | 10 | 2exbii 1583 | . . 3 ⊢ (∃x∃y∃z(z = 〈x, y〉 ∧ φ) ↔ ∃x∃yφ) |
12 | 4, 11 | bitri 240 | . 2 ⊢ (∃z∃x∃y(z = 〈x, y〉 ∧ φ) ↔ ∃x∃yφ) |
13 | 1, 3, 12 | 3bitri 262 | 1 ⊢ ({〈x, y〉 ∣ φ} ≠ ∅ ↔ ∃x∃yφ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 〈cop 4561 {copab 4622 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: (None) |
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