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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 3045 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssuni 3681 | . 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | mpan 416 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 ⊆ wss 3000 ∪ cuni 3659 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-uni 3660 |
This theorem is referenced by: unissel 3688 ssunieq 3692 pwuni 4033 pwel 4054 uniopel 4092 iunpw 4315 dmrnssfld 4709 fvssunirng 5333 relfvssunirn 5334 sefvex 5339 riotaexg 5626 pwuninel2 6061 tfrlem9 6098 tfrexlem 6113 sbthlem1 6720 sbthlem2 6721 unirnioo 9452 eltopss 11769 toponss 11785 isbasis3g 11805 baspartn 11809 bastg 11822 tgcl 11825 epttop 11851 difopn 11869 ssntr 11883 isopn3 11886 isopn3i 11896 bj-elssuniab 11964 |
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