Theorem sn-iotaex 40197

Description: iotaex 6413 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
sn-iotaex	⊢ (℩𝑥𝜑) ∈ V

Proof of Theorem sn-iotaex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotavallem 40192	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2847	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1933	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		sn-iotanul 40194	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5231	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2847	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  Vcvv 3432  ∅c0 4256  {csn 4561  ℩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  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  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-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391

This theorem is referenced by: (None)