Theorem cnv0OLD 5872

Description: Obsolete version of cnv0 5871 as of 31-Jan-2026. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5259, ax-nul 5271, ax-pr 5406. (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 5162	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 492	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1833	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1833	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5671	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5176	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
7	5, 6	eqtri 2788	. . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
8	7	eqabri 2907	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	4, 8	mtbir 326	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
10	9	nel0 4309	1 ⊢ ◡∅ = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∅c0 4286  ⟨cop 4597   class class class wbr 5111  {copab 5175  ^◡ccnv 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-dif 3909  df-nul 4287  df-br 5112  df-opab 5176  df-cnv 5671

This theorem is used by: (None)