Theorem cnv0OLD 5870

Description: Obsolete version of cnv0 5869 as of 31-Jan-2026. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5257, ax-nul 5269, ax-pr 5404. (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 5160	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 491	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1830	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1830	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5669	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5174	. . . . 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 4309	1 ⊢ ◡∅ = ∅

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∅c0 4286  ⟨cop 4595   class class class wbr 5109  {copab 5173  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-dif 3908  df-nul 4287  df-br 5110  df-opab 5174  df-cnv 5669

This theorem is referenced by: (None)