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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 7099 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
2 | ssrab2 4007 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | 2 | unissi 4809 | . . . 4 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ ∪ 𝐴 |
4 | ssun2 4100 | . . . 4 ⊢ ∪ 𝐴 ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
5 | 3, 4 | sstri 3924 | . . 3 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
6 | 1, 5 | eqsstrdi 3969 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
7 | riotaund 7132 | . . 3 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) | |
8 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) | |
9 | 7, 8 | eqsstrdi 3969 | . 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)) |
10 | 6, 9 | pm2.61i 185 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃!wreu 3108 {crab 3110 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 ℩crio 7092 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 df-riota 7093 |
This theorem is referenced by: (None) |
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