Theorem bj-ab0 32902

Description: The class of sets verifying a falsity is the empty set (closed form of abf 3978). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ab0	⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Proof of Theorem bj-ab0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1839	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑)
2		bj-stdpc4v 32754	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3		sbn 2391	. . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 208	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5		df-clab 2609	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
6	4, 5	sylnibr 319	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
7	1, 6	alrimih 1751	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
8		eq0 3929	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
9	7, 8	sylibr 224	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481   = wceq 1483  [wsb 1880   ∈ wcel 1990  {cab 2608  ∅c0 3915

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  bj-abf  32903  bj-csbprc  32904