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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 6324 | . . 3 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | eqimss 4022 | . . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) |
4 | iotanul 6327 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
5 | 0ss 4349 | . . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑} | |
6 | 4, 5 | eqsstrdi 4020 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}) |
7 | 3, 6 | pm2.61i 184 | 1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∃!weu 2649 {cab 2799 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 ℩cio 6306 |
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 2157 ax-12 2173 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-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-pr 4563 df-uni 4832 df-iota 6308 |
This theorem is referenced by: bj-nuliotaALT 34345 |
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