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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 4200 | . 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 3961 ⊆ wss 3963 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 |
This theorem is referenced by: pwunss 4623 dmrnssfld 5987 tc2 9780 djuunxp 9959 pwxpndom2 10703 ltrelxr 11320 nn0ssre 12528 nn0sscn 12529 nn0ssz 12634 dfle2 13186 difreicc 13521 hashxrcl 14393 ramxrcl 17051 strleun 17191 cssincl 21724 leordtval2 23236 lecldbas 23243 comppfsc 23556 aalioulem2 26390 taylfval 26415 addsbdaylem 28064 addsbday 28065 addsdilem3 28194 addsdilem4 28195 mulsasslem3 28206 pw2bday 28433 axlowdimlem10 28981 shunssji 31398 shsval3i 31417 shjshsi 31521 spanuni 31573 sshhococi 31575 esumcst 34044 hashf2 34065 sxbrsigalem3 34254 signswch 34555 bj-unrab 36909 bj-tagss 36963 bj-imdirco 37173 poimirlem16 37623 poimirlem19 37626 poimirlem23 37630 poimirlem29 37636 poimirlem31 37638 poimirlem32 37639 mblfinlem3 37646 mblfinlem4 37647 hdmapevec 41818 rtrclex 43607 trclexi 43610 rtrclexi 43611 cnvrcl0 43615 cnvtrcl0 43616 comptiunov2i 43696 cotrclrcl 43732 cncfiooicclem1 45849 fourierdlem62 46124 |
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