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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 3167 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssuni 3816 | . 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | mpan 422 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ⊆ wss 3121 ∪ cuni 3794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-uni 3795 |
This theorem is referenced by: unissel 3823 ssunieq 3827 pwuni 4176 pwel 4201 uniopel 4239 iunpw 4463 dmrnssfld 4872 fvssunirng 5509 relfvssunirn 5510 sefvex 5515 riotaexg 5811 pwuninel2 6259 tfrlem9 6296 tfrexlem 6311 sbthlem1 6931 sbthlem2 6932 unirnioo 9919 eltopss 12762 toponss 12779 isbasis3g 12799 baspartn 12803 bastg 12816 tgcl 12819 epttop 12845 difopn 12863 ssntr 12877 isopn3 12880 isopn3i 12890 neiuni 12916 resttopon 12926 restopn2 12938 ssidcn 12965 lmtopcnp 13005 txuni2 13011 hmeoimaf1o 13069 tgioo 13301 bj-elssuniab 13787 |
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