Theorem iotanul2 6531

Description: Version of iotanul 6539 using df-iota 6514 instead of dfiota2 6515. (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 6514	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		n0 4353	. . . 4 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}})
3		eluni 4910	. . . . . 6 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}))
4		vex 3484	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5		sneq 4636	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	elab 3679	. . . . . . . . 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 1835	. . . . . 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 1930	. . . 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 2969	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = ∅)
15	1, 14	eqtrid 2789	1 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∅c0 4333  {csn 4626  ∪ cuni 4907  ℩cio 6512

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627  df-uni 4908  df-iota 6514

This theorem is referenced by:  iotassuni  6533  iotaex  6534  sn-tz6.12-2  42690