Theorem iotassuni 6500

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 2178, ax-11 2194, ax-12 2215. (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 6497	. . 3 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		eqimss 3997	. . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl 18	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
4		iotanul2 6498	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
5		0ss 4357	. . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑}
6	4, 5	eqsstrdi 3983	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
7	3, 6	pm2.61i 184	1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1563  ∃wex 1802  {cab 2743   ⊆ wss 3907  ∅c0 4288  {csn 4585  ∪ cuni 4867  ℩cio 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-uni 4868  df-iota 6481

This theorem is referenced by:  bj-nuliotaALT  37550