Theorem notzfaus 5110

Description: In the Separation Scheme zfauscl 5056, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction if that requirement is not met. (Contributed by NM, 8-Feb-2006.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem notzfaus

Step	Hyp	Ref	Expression
1		notzfaus.1	. . . . . 6 ⊢ 𝐴 = {∅}
2		0ex 5062	. . . . . . 7 ⊢ ∅ ∈ V
3	2	snnz 4579	. . . . . 6 ⊢ {∅} ≠ ∅
4	1, 3	eqnetri 3031	. . . . 5 ⊢ 𝐴 ≠ ∅
5		n0 4190	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
6	4, 5	mpbi 222	. . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴
7		biimt 353	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
8		iman 393	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦))
9		notzfaus.2	. . . . . . . 8 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)
10	9	anbi2i 613	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦))
11	8, 10	xchbinxr 327	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
12	7, 11	syl6bb 279	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
13		xor3 375	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
14	12, 13	sylibr 226	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	6, 14	eximii 1799	. . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
16		exnal 1789	. . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
17	15, 16	mpbi 222	. 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
18	17	nex 1763	1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507  ∃wex 1742   ∈ wcel 2050   ≠ wne 2961  ∅c0 4172  {csn 4435

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  ax-nul 5061

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-ne 2962  df-v 3411  df-dif 3826  df-nul 4173  df-sn 4436

This theorem is referenced by: (None)