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Mirrors > Home > MPE Home > Th. List > unssi | Structured version Visualization version GIF version |
Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
unssi.1 | ⊢ 𝐴 ⊆ 𝐶 |
unssi.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
unssi | ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unssi.1 | . . 3 ⊢ 𝐴 ⊆ 𝐶 | |
2 | unssi.2 | . . 3 ⊢ 𝐵 ⊆ 𝐶 | |
3 | 1, 2 | pm3.2i 470 | . 2 ⊢ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) |
4 | unss 4213 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
5 | 3, 4 | mpbi 230 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∪ cun 3974 ⊆ wss 3976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-v 3490 df-un 3981 df-ss 3993 |
This theorem is referenced by: pwunss 4640 dmrnssfld 5996 tc2 9811 djuunxp 9990 pwxpndom2 10734 ltrelxr 11351 nn0ssre 12557 nn0sscn 12558 nn0ssz 12662 dfle2 13209 difreicc 13544 hashxrcl 14406 ramxrcl 17064 strleun 17204 cssincl 21729 leordtval2 23241 lecldbas 23248 comppfsc 23561 aalioulem2 26393 taylfval 26418 addsbdaylem 28067 addsbday 28068 addsdilem3 28197 addsdilem4 28198 mulsasslem3 28209 pw2bday 28436 axlowdimlem10 28984 shunssji 31401 shsval3i 31420 shjshsi 31524 spanuni 31576 sshhococi 31578 esumcst 34027 hashf2 34048 sxbrsigalem3 34237 signswch 34538 bj-unrab 36892 bj-tagss 36946 bj-imdirco 37156 poimirlem16 37596 poimirlem19 37599 poimirlem23 37603 poimirlem29 37609 poimirlem31 37611 poimirlem32 37612 mblfinlem3 37619 mblfinlem4 37620 hdmapevec 41792 rtrclex 43579 trclexi 43582 rtrclexi 43583 cnvrcl0 43587 cnvtrcl0 43588 comptiunov2i 43668 cotrclrcl 43704 cncfiooicclem1 45814 fourierdlem62 46089 |
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