Theorem bj-ab0 33716

Description: The class of sets verifying a falsity is the empty set (closed form of abf 4236). (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 1869	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑)
2		stdpc4 2019	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3		sbn 2214	. . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 210	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5		df-clab 2753	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
6	4, 5	sylnibr 321	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
7	1, 6	alrimih 1786	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
8		eq0 4188	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
9	7, 8	sylibr 226	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505   = wceq 1507  [wsb 2015   ∈ wcel 2050  {cab 2752  ∅c0 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-dif 3826  df-nul 4173

This theorem is referenced by:  bj-abf  33717  bj-csbprc  33718