Theorem iotaex

Description: Theorem 8.23 in Quine p. 58. This theorem proves the existence of the iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011) Remove dependency on ax-10 , ax-11 , ax-12 . (Revised by SN, 6-Nov-2024)

	Ref	Expression
Assertion	iotaex	⊢ ( ℩ 𝑥 𝜑 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		iotaval2	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = 𝑦 )
2		vex	⊢ 𝑦 ∈ V
3	1 2	eqeltrdi	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) ∈ V )
4	3	exlimiv	⊢ ( ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) ∈ V )
5		iotanul2	⊢ ( ¬ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = ∅ )
6		0ex	⊢ ∅ ∈ V
7	5 6	eqeltrdi	⊢ ( ¬ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) ∈ V )
8	4 7	pm2.61i	⊢ ( ℩ 𝑥 𝜑 ) ∈ V