Theorem opabidw 5530

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5531 with a disjoint variable condition, which does not require ax-13 2366. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2366. (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
opabidw	⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabidw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5470	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5496	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 222	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5216	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3670	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ⟨cop 4639  {copab 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-opab 5216

This theorem is referenced by:  rexopabb  5534  ssopab2bw  5553  dmopab  5922  rnopab  5960  funopab  6594  opabiota  6985  fvopab5  7042  f1ompt  7125  ovid  7567  zfrep6  7968  enssdom  9008  omxpenlem  9111  infxpenlem  10056  canthwelem  10693  pospo  18370  2ndcdisj  23451  lgsquadlem1  27409  lgsquadlem2  27410  h2hlm  30913  opabdm  32531  opabrn  32532  fpwrelmap  32647  eulerpartlemgvv  34210  fineqvrep  34931  satfvsucsuc  35193  bj-opelopabid  36894  phpreu  37305  poimirlem26  37347  vvdifopab  37958  brabidgaw  38063  diclspsn  40893  areaquad  42881  sprsymrelf  47067