Theorem notzfausOLD 5256

Description: Obsolete proof of notzfaus 5255 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem notzfausOLD

Step	Hyp	Ref	Expression
1		notzfaus.1	. . . . . 6 ⊢ 𝐴 = {∅}
2		0ex 5204	. . . . . . 7 ⊢ ∅ ∈ V
3	2	snnz 4704	. . . . . 6 ⊢ {∅} ≠ ∅
4	1, 3	eqnetri 3085	. . . . 5 ⊢ 𝐴 ≠ ∅
5		n0 4303	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
6	4, 5	mpbi 232	. . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴
7		biimt 363	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦)))
8		iman 404	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦))
9		notzfaus.2	. . . . . . . 8 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)
10	9	anbi2i 624	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦))
11	8, 10	xchbinxr 337	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
12	7, 11	syl6bb 289	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
13		xor3 386	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
14	12, 13	sylibr 236	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	6, 14	eximii 1836	. . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
16		exnal 1826	. . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
17	15, 16	mpbi 232	. 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
18	17	nex 1800	1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ≠ wne 3015  ∅c0 4284  {csn 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-nul 5203

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-v 3493  df-dif 3932  df-nul 4285  df-sn 4561

This theorem is referenced by: (None)