Theorem bj-vn0ALT 37708

Description: Alternate proof of vn0 4298 which does not use eqabbw 2836 (and is shorter than vn0 4298 when eqabbw 2836 is inlined). (Contributed by BJ, 12-Jul-2026.) Using the same dummy variable for 𝑦 and 𝑧 slightly reduces the proof size. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-vn0ALT	⊢ V ≠ ∅

Proof of Theorem bj-vn0ALT

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fal 1584	. . 3 ⊢ ¬ ⊥
2		dfv2 3458	. . . . 5 ⊢ V = {𝑦 ∣ ⊤}
3		dfnul4 4288	. . . . 5 ⊢ ∅ = {𝑧 ∣ ⊥}
4	2, 3	eqeq12i 2781	. . . 4 ⊢ (V = ∅ ↔ {𝑦 ∣ ⊤} = {𝑧 ∣ ⊥})
5		dfcleq 2756	. . . . 5 ⊢ ({𝑦 ∣ ⊤} = {𝑧 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ ⊤} ↔ 𝑥 ∈ {𝑧 ∣ ⊥}))
6		df-clab 2742	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑦 ∣ ⊤} ↔ [𝑥 / 𝑦]⊤)
7		sbv 2122	. . . . . . . . 9 ⊢ ([𝑥 / 𝑦]⊤ ↔ ⊤)
8	6, 7	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∣ ⊤} ↔ ⊤)
9		df-clab 2742	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑧 ∣ ⊥} ↔ [𝑥 / 𝑧]⊥)
10		sbv 2122	. . . . . . . . 9 ⊢ ([𝑥 / 𝑧]⊥ ↔ ⊥)
11	9, 10	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ ⊥} ↔ ⊥)
12	8, 11	bibi12i 342	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑦 ∣ ⊤} ↔ 𝑥 ∈ {𝑧 ∣ ⊥}) ↔ (⊤ ↔ ⊥))
13		trubifal 1601	. . . . . . 7 ⊢ ((⊤ ↔ ⊥) ↔ ⊥)
14	12, 13	sylbb 222	. . . . . 6 ⊢ ((𝑥 ∈ {𝑦 ∣ ⊤} ↔ 𝑥 ∈ {𝑧 ∣ ⊥}) → ⊥)
15	14	spsv 2017	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ ⊤} ↔ 𝑥 ∈ {𝑧 ∣ ⊥}) → ⊥)
16	5, 15	sylbi 220	. . . 4 ⊢ ({𝑦 ∣ ⊤} = {𝑧 ∣ ⊥} → ⊥)
17	4, 16	sylbi 220	. . 3 ⊢ (V = ∅ → ⊥)
18	1, 17	mto 200	. 2 ⊢ ¬ V = ∅
19	18	neir 2961	1 ⊢ V ≠ ∅

Syntax hints:   ↔ wb 209  ∀wal 1568   = wceq 1570  ⊤wtru 1571  ⊥wfal 1582  [wsb 2096   ∈ wcel 2143  {cab 2741   ≠ wne 2958  Vcvv 3455  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ne 2959  df-v 3457  df-dif 3908  df-nul 4287

This theorem is referenced by: (None)