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Mirrors > Home > MPE Home > Th. List > unss | Structured version Visualization version GIF version |
Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.) |
Ref | Expression |
---|---|
unss | ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3845 | . 2 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶)) | |
2 | 19.26 1833 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) | |
3 | elun 4013 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
4 | 3 | imbi1i 342 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
5 | jaob 944 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) | |
6 | 4, 5 | bitri 267 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
7 | 6 | albii 1782 | . . 3 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
8 | dfss2 3845 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
9 | dfss2 3845 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) | |
10 | 8, 9 | anbi12i 617 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
11 | 2, 7, 10 | 3bitr4i 295 | . 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
12 | 1, 11 | bitr2i 268 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 ∀wal 1505 ∈ wcel 2048 ∪ cun 3826 ⊆ wss 3828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2747 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-v 3414 df-un 3833 df-in 3835 df-ss 3842 |
This theorem is referenced by: unssi 4048 unssd 4049 unssad 4050 unssbd 4051 nsspssun 4120 uneqin 4141 prssg 4624 ssunsn2 4632 tpss 4640 iunopeqop 5264 pwundif 5306 eqrelrel 5517 xpsspw 5529 relun 5531 relcoi2 5964 fnsuppres 7657 wfrlem15 7770 dfer2 8086 isinf 8522 fiin 8677 trcl 8960 supxrun 12522 trclun 14229 isumltss 15057 rpnnen2lem12 15432 lcmfunsnlem 15835 lcmfun 15839 coprmprod 15855 coprmproddvdslem 15856 lubun 17585 isipodrs 17623 fpwipodrs 17626 ipodrsima 17627 aspval2 19835 unocv 20520 uncld 21347 restntr 21488 cmpcld 21708 uncmp 21709 ufprim 22215 tsmsfbas 22433 ovolctb2 23790 ovolun 23797 unmbl 23835 plyun0 24484 sshjcl 28907 sshjval2 28963 shlub 28966 ssjo 28999 spanuni 29096 dfon2lem3 32520 dfon2lem7 32524 noextendseq 32665 noresle 32691 clsun 33167 lindsadd 34304 lindsenlbs 34306 mblfinlem3 34350 ismblfin 34352 paddssat 36373 pclunN 36457 paddunN 36486 poldmj1N 36487 pclfinclN 36509 lsmfgcl 39048 ssuncl 39269 sssymdifcl 39271 undmrnresiss 39304 mptrcllem 39314 cnvrcl0 39326 dfrtrcl5 39330 brtrclfv2 39413 unhe1 39472 dffrege76 39626 uneqsn 39714 clsk1indlem3 39734 mnurndlem1 39970 setrec1lem4 44134 elpglem2 44155 |
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