Theorem opabidw 5524

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5525 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2370. (Revised by Gino Giotto, 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 5464	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5490	. . 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 5211	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3672	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ⟨cop 4634  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211

This theorem is referenced by:  rexopabb  5528  ssopab2bw  5547  dmopab  5915  rnopab  5953  funopab  6583  opabiota  6974  fvopab5  7030  f1ompt  7112  ovid  7552  zfrep6  7944  enssdom  8976  omxpenlem  9076  infxpenlem  10011  canthwelem  10648  pospo  18303  2ndcdisj  23181  lgsquadlem1  27120  lgsquadlem2  27121  h2hlm  30501  opabdm  32108  opabrn  32109  fpwrelmap  32226  eulerpartlemgvv  33674  fineqvrep  34394  satfvsucsuc  34655  bj-opelopabid  36372  phpreu  36776  poimirlem26  36818  vvdifopab  37432  brabidgaw  37538  diclspsn  40369  areaquad  42268  sprsymrelf  46462