Theorem iotassuni 5784

Description: The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

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

Proof of Theorem iotassuni

Step	Hyp	Ref	Expression
1		iotauni 5780	. . 3 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		eqimss 3620	. . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl 17	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
4		iotanul 5783	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5		0ss 3924	. . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑}
6	4, 5	syl6eqss 3618	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
7	3, 6	pm2.61i 175	1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1475  ∃!weu 2458  {cab 2596   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768

This theorem is referenced by:  bj-nuliotaALT  32211