Theorem iotassuni 6532

Description: The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) Remove dependency on ax-10 2140, ax-11 2156, ax-12 2176. (Revised by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotassuni	⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Proof of Theorem iotassuni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotauni2 6529	. . 3 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		eqimss 4041	. . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl 17	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
4		iotanul2 6530	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
5		0ss 4399	. . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑}
6	4, 5	eqsstrdi 4027	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
7	3, 6	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1539  ∃wex 1778  {cab 2713   ⊆ wss 3950  ∅c0 4332  {csn 4625  ∪ cuni 4906  ℩cio 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513

This theorem is referenced by:  bj-nuliotaALT  37060