Theorem riotassuni 7358

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 7324	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
2		ssrab2 4021	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3	2	unissi 4860	. . . 4 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ ∪ 𝐴
4		ssun2 4120	. . . 4 ⊢ ∪ 𝐴 ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
5	3, 4	sstri 3932	. . 3 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
6	1, 5	eqsstrdi 3967	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴))
7		riotaund 7357	. . 3 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)
8		0ss 4341	. . 3 ⊢ ∅ ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
9	7, 8	eqsstrdi 3967	. 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 3341  {crab 3390   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ℩crio 7317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-reu 3344  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-sn 4569  df-pr 4571  df-uni 4852  df-iota 6449  df-riota 7318

This theorem is referenced by: (None)