Theorem riotassuni 7384

Description: The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotassuni	⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotassuni

Step	Hyp	Ref	Expression
1		riotauni 7350	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
2		ssrab2 4043	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3	2	unissi 4880	. . . 4 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ ∪ 𝐴
4		ssun2 4142	. . . 4 ⊢ ∪ 𝐴 ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
5	3, 4	sstri 3956	. . 3 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
6	1, 5	eqsstrdi 3991	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴))
7		riotaund 7383	. . 3 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)
8		0ss 4363	. . 3 ⊢ ∅ ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
9	7, 8	eqsstrdi 3991	. 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴))
10	6, 9	pm2.61i 182	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)

Syntax hints:  ¬ wn 3  ∃!wreu 3352  {crab 3405   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  ∪ cuni 4871  ℩crio 7343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-sn 4590  df-pr 4592  df-uni 4872  df-iota 6464  df-riota 7344

This theorem is referenced by: (None)