Theorem sn-iotanul 40194

Description: Version of iotanul 6411 using df-iota 6391 instead of dfiota2 6392. (Contributed by SN, 6-Nov-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotanul

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

Step	Hyp	Ref	Expression
1		df-iota 6391	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		n0 4280	. . . 4 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}})
3		eluni 4842	. . . . . 6 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}))
4		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5		sneq 4571	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
6	5	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	4, 6	elab 3609	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ↔ {𝑥 ∣ 𝜑} = {𝑦})
8	7	biimpi 215	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → {𝑥 ∣ 𝜑} = {𝑦})
9	8	adantl 482	. . . . . . 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 216	. . . . 5 ⊢ (𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
12	11	exlimiv 1933	. . . 4 ⊢ (∃𝑣 𝑣 ∈ ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
13	2, 12	sylbi 216	. . 3 ⊢ (∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
14	13	necon1bi 2972	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = ∅)
15	1, 14	eqtrid 2790	1 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∅c0 4256  {csn 4561  ∪ cuni 4839  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-nul 4257  df-sn 4562  df-uni 4840  df-iota 6391

This theorem is referenced by:  sn-iotassuni  40196  sn-iotaex  40197