Theorem opabf 34131

Description: A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.)

Hypothesis

Ref	Expression
opabf.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
opabf	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabf

Step	Hyp	Ref	Expression
1		rel0 5243	. 2 ⊢ Rel ∅
2		opabf.1	. . . 4 ⊢ ¬ 𝜑
3	2	pm2.21i 116	. . 3 ⊢ (𝜑 → ⟨𝑥, 𝑦⟩ ∈ ∅)
4	3	opabssi 34130	. 2 ⊢ (Rel ∅ → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ ∅)
5		ss0 3974	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ ∅ → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
6	1, 4, 5	mp2b 10	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183  {copab 4712  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)