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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 7312 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
2 | ssrab2 4036 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | 2 | unissi 4873 | . . . 4 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ ∪ 𝐴 |
4 | ssun2 4132 | . . . 4 ⊢ ∪ 𝐴 ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
5 | 3, 4 | sstri 3952 | . . 3 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
6 | 1, 5 | eqsstrdi 3997 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
7 | riotaund 7346 | . . 3 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) | |
8 | 0ss 4355 | . . 3 ⊢ ∅ ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
9 | 7, 8 | eqsstrdi 3997 | . 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
10 | 6, 9 | pm2.61i 182 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃!wreu 3350 {crab 3406 ∪ cun 3907 ⊆ wss 3909 ∅c0 4281 𝒫 cpw 4559 ∪ cuni 4864 ℩crio 7305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-sn 4586 df-pr 4588 df-uni 4865 df-iota 6444 df-riota 7306 |
This theorem is referenced by: (None) |
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