Theorem sn-iotaex 40095

Description: iotaex 6395 without ax-10 2143, ax-11 2160, ax-12 2177. (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		sn-iotaval 40091	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3427	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2848	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1938	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		sn-iotanul 40093	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5224	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2848	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 185	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   = wceq 1543  ∃wex 1787   ∈ wcel 2112  {cab 2716  Vcvv 3423  ∅c0 4254  {csn 4558  ℩cio 6371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-nul 5223

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-v 3425  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6373

This theorem is referenced by: (None)