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Mirrors > Home > MPE Home > Th. List > iotassuni | Structured version Visualization version GIF version |
Description: The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
iotassuni | ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotauni 6333 | . . 3 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | eqimss 3943 | . . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) |
4 | iotanul 6336 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
5 | 0ss 4297 | . . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑} | |
6 | 4, 5 | eqsstrdi 3941 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) |
7 | 3, 6 | pm2.61i 185 | 1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∃!weu 2567 {cab 2714 ⊆ wss 3853 ∅c0 4223 ∪ cuni 4805 ℩cio 6314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-pr 4530 df-uni 4806 df-iota 6316 |
This theorem is referenced by: bj-nuliotaALT 34915 |
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