Theorem iotanul2 6498

Description: Version of iotanul 6505 using df-iota 6481 instead of dfiota2 6482. (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 6481	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		n0 4308	. . . 4 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}})
3		eluni 4870	. . . . . 6 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}))
4		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4595	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	elab 3641	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ {𝑥 ∣ 𝜑} = {𝑦})
8	7	bilani 509	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}) → {𝑥 ∣ 𝜑} = {𝑦})
9	8	eximi 1858	. . . . . 6 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
10	3, 9	sylbi 220	. . . . 5 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
11	10	exlimiv 1953	. . . 4 ⊢ (∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
12	2, 11	sylbi 220	. . 3 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
13	12	necon1bi 2988	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = ∅)
14	1, 13	eqtrid 2812	1 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∅c0 4288  {csn 4585  ∪ cuni 4867  ℩cio 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-nul 4289  df-sn 4586  df-uni 4868  df-iota 6481

This theorem is referenced by:  iotassuni  6500  iotaex  6501  tz6.12-2  6858