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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 3591 | . 2 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶)) | |
2 | 19.26 1798 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) | |
3 | elun 3753 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
4 | 3 | imbi1i 339 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
5 | jaob 822 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) | |
6 | 4, 5 | bitri 264 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
7 | 6 | albii 1747 | . . 3 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
8 | dfss2 3591 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
9 | dfss2 3591 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) | |
10 | 8, 9 | anbi12i 733 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))) |
11 | 2, 7, 10 | 3bitr4i 292 | . 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
12 | 1, 11 | bitr2i 265 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 |
This theorem is referenced by: unssi 3788 unssd 3789 unssad 3790 unssbd 3791 nsspssun 3857 uneqin 3878 uneqdifeqOLD 4058 prssg 4350 prssOLD 4352 ssunsn2 4359 tpss 4368 iunopeqop 4981 pwundif 5021 eqrelrel 5221 xpsspw 5233 relun 5235 relcoi2 5663 fnsuppres 7322 wfrlem15 7429 dfer2 7743 isinf 8173 fiin 8328 trcl 8604 supxrun 12146 trclun 13755 isumltss 14580 rpnnen2lem12 14954 lcmfunsnlem 15354 lcmfun 15358 coprmprod 15375 coprmproddvdslem 15376 lubun 17123 isipodrs 17161 fpwipodrs 17164 ipodrsima 17165 aspval2 19347 unocv 20024 uncld 20845 restntr 20986 cmpcld 21205 uncmp 21206 ufprim 21713 tsmsfbas 21931 ovolctb2 23260 ovolun 23267 unmbl 23305 plyun0 23953 sshjcl 28214 sshjval2 28270 shlub 28273 ssjo 28306 spanuni 28403 dfon2lem3 31690 dfon2lem7 31694 noextendseq 31820 noresle 31846 clsun 32323 lindsenlbs 33404 mblfinlem3 33448 ismblfin 33450 paddssat 35100 pclunN 35184 paddunN 35213 poldmj1N 35214 pclfinclN 35236 lsmfgcl 37644 ssuncl 37875 sssymdifcl 37877 undmrnresiss 37910 mptrcllem 37920 cnvrcl0 37932 dfrtrcl5 37936 brtrclfv2 38019 unhe1 38079 dffrege76 38233 uneqsn 38321 clsk1indlem3 38341 setrec1lem4 42437 elpglem2 42455 |
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