Theorem iotanul2 6516

Description: Version of iotanul 6523 using df-iota 6499 instead of dfiota2 6500. (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 6499	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		n0 4310	. . . 4 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}})
3		eluni 4880	. . . . . 6 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}))
4		vex 3462	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5		sneq 4604	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2777	. . . . . . . . 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 510	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}) → {𝑥 ∣ 𝜑} = {𝑦})
9	8	eximi 1868	. . . . . 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 1963	. . . 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 2989	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = ∅)
14	1, 13	eqtrid 2813	1 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∅c0 4289  {csn 4594  ∪ cuni 4877  ℩cio 6497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-v 3460  df-dif 3911  df-nul 4290  df-sn 4595  df-uni 4878  df-iota 6499

This theorem is used by:  iotassuni  6518  iotaex  6519  tz6.12-2  6875