Theorem iotanul2 6473

Description: Version of iotanul 6480 using df-iota 6456 instead of dfiota2 6457. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotanul2	⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotanul2

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 6456	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		n0 4307	. . . 4 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}})
3		eluni 4868	. . . . . 6 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}))
4		vex 3446	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5		sneq 4592	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	elab 3636	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ {𝑥 ∣ 𝜑} = {𝑦})
8	7	biimpi 216	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → {𝑥 ∣ 𝜑} = {𝑦})
9	8	adantl 481	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}) → {𝑥 ∣ 𝜑} = {𝑦})
10	9	eximi 1837	. . . . . 6 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
11	3, 10	sylbi 217	. . . . 5 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
12	11	exlimiv 1932	. . . 4 ⊢ (∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
13	2, 12	sylbi 217	. . 3 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
14	13	necon1bi 2961	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = ∅)
15	1, 14	eqtrid 2784	1 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∅c0 4287  {csn 4582  ∪ cuni 4865  ℩cio 6454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-nul 4288  df-sn 4583  df-uni 4866  df-iota 6456

This theorem is referenced by:  iotassuni  6475  iotaex  6476  tz6.12-2  6829