Theorem iotassuni 6461

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 2146, ax-11 2162, ax-12 2182. (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 6458	. . 3 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		eqimss 3989	. . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl 17	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
4		iotanul2 6459	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
5		0ss 4349	. . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑}
6	4, 5	eqsstrdi 3975	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
7	3, 6	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1541  ∃wex 1780  {cab 2711   ⊆ wss 3898  ∅c0 4282  {csn 4575  ∪ cuni 4858  ℩cio 6440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859  df-iota 6442

This theorem is referenced by:  bj-nuliotaALT  37123