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Mirrors > Home > MPE Home > Th. List > riotassuni | Structured version Visualization version GIF version |
Description: The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
riotassuni | ⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotauni 7114 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
2 | ssrab2 4056 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | 2 | unissi 4855 | . . . 4 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ ∪ 𝐴 |
4 | ssun2 4149 | . . . 4 ⊢ ∪ 𝐴 ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
5 | 3, 4 | sstri 3976 | . . 3 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
6 | 1, 5 | eqsstrdi 4021 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
7 | riotaund 7147 | . . 3 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) | |
8 | 0ss 4350 | . . 3 ⊢ ∅ ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
9 | 7, 8 | eqsstrdi 4021 | . 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
10 | 6, 9 | pm2.61i 184 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃!wreu 3140 {crab 3142 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4832 ℩crio 7107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4562 df-pr 4564 df-uni 4833 df-iota 6309 df-riota 7108 |
This theorem is referenced by: (None) |
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