Theorem cnv0OLD 5866

Description: Obsolete version of cnv0 5865 as of 31-Jan-2026. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5251, ax-nul 5262, ax-pr 5394. (Revised by KP, 25-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnv0OLD	⊢ ◡∅ = ∅

Proof of Theorem cnv0OLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		br0 5156	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 492	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1832	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1832	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5662	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5170	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
7	5, 6	eqtri 2786	. . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
8	7	eqabri 2905	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	4, 8	mtbir 326	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
10	9	nel0 4304	1 ⊢ ◡∅ = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146  {cab 2741  ∅c0 4280  ⟨cop 4837   class class class wbr 5105  {copab 5169  ^◡ccnv 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-dif 4049  df-nul 4281  df-br 5106  df-opab 5170  df-cnv 5662

This theorem is used by: (None)