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Mirrors > Home > ILE Home > Th. List > elssuni | GIF version |
Description: An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
elssuni | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3190 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssuni 3846 | . 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ⊆ wss 3144 ∪ cuni 3824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-uni 3825 |
This theorem is referenced by: unissel 3853 ssunieq 3857 pwuni 4210 pwel 4236 uniopel 4274 iunpw 4498 dmrnssfld 4908 iotaexab 5214 fvssunirng 5549 relfvssunirn 5550 sefvex 5555 riotaexg 5855 pwuninel2 6306 tfrlem9 6343 tfrexlem 6358 sbthlem1 6985 sbthlem2 6986 unirnioo 10002 eltopss 13961 toponss 13978 isbasis3g 13998 baspartn 14002 bastg 14013 tgcl 14016 epttop 14042 difopn 14060 ssntr 14074 isopn3 14077 isopn3i 14087 neiuni 14113 resttopon 14123 restopn2 14135 ssidcn 14162 lmtopcnp 14202 txuni2 14208 hmeoimaf1o 14266 tgioo 14498 bj-elssuniab 14996 |
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