Theorem opabn0 5539

Description: Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)

Assertion

Ref	Expression
opabn0	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑)

Proof of Theorem opabn0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4315	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2		elopab 5512	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3	2	exbii 1875	. . 3 ⊢ (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		exrot3 2206	. . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		opex 5446	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
6	5	isseti 3481	. . . . . 6 ⊢ ∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩
7		19.41v 1976	. . . . . 6 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	6, 7	mpbiran 721	. . . . 5 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
9	8	2exbii 1876	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
10	4, 9	bitri 278	. . 3 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
11	3, 10	bitri 278	. 2 ⊢ (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑)
12	1, 11	bitri 278	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∅c0 4294  ⟨cop 4600  {copab 5177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178

This theorem is referenced by:  opab0  5540  csbopab  5541  dvdsrval  20442  thlle  21815  bcthlem5  25455  lgsquadlem3  27511  disjecxrn  38950  br1cosscnvxrn  39102