Theorem notsep 5334

Description: In the Separation Scheme sepgi 5259, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). That requirement is necessary: notsep 5334, which requires only ax-ext 2733 and ax-nul 5268 on top of first-order logic, derives *non*-existence of a "separating set" for specific values of the containing set 𝐴 and of the separating condition 𝜑. Therefore, from the axioms required by notsep 5334 and an overly strong axiom of separation without the requirement that 𝑦 not occur in 𝜑, we could derive ⊥, a contradiction. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.)

Hypotheses

Ref	Expression
notsep.1	⊢ 𝐴 = {∅}
notsep.2	⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)

Assertion

Ref	Expression
notsep	⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem notsep

Step	Hyp	Ref	Expression
1		notsep.1	. . . . . 6 ⊢ 𝐴 = {∅}
2		0ex 5269	. . . . . . 7 ⊢ ∅ ∈ V
3	2	snnz 4741	. . . . . 6 ⊢ {∅} ≠ ∅
4	1, 3	eqnetri 3026	. . . . 5 ⊢ 𝐴 ≠ ∅
5		n0 4306	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
6	4, 5	mpbi 233	. . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴
7		pm5.19 390	. . . . 5 ⊢ ¬ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)
8		ibar 537	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
9		notsep.2	. . . . . . 7 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)
10	8, 9	bitr3di 289	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ 𝑥 ∈ 𝑦))
11	10	bibi2d 345	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦)))
12	7, 11	mtbiri 330	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
13	6, 12	eximii 1865	. . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
14		exnal 1855	. . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	13, 14	mpbi 233	. 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
16	15	nex 1828	1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  ∅c0 4285  {csn 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-nul 4286  df-sn 4589

This theorem is referenced by: (None)