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Theorem basgen 14259

Description: Given a topology

, show that a subset

satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

**Assertion**
Ref	Expression
basgen	$/\$ $/\$

**Proof of Theorem basgen**
Step	Hyp	Ref	Expression
1		tgss 14242	. . . 4 $/\$
2	1	3adant3 1019	. . 3 $/\$ $/\$
3		tgtop 14247	. . . 4
4	3	3ad2ant1 1020	. . 3 $/\$ $/\$
5	2, 4	sseqtrd 3218	. 2 $/\$ $/\$
6		simp3 1001	. 2 $/\$ $/\$
7	5, 6	eqssd 3197	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ w3a 980

wceq 1364

wcel 2164

wss 3154

cfv 5255

ctg 12868

ctop 14176

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-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2987 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-topgen 12874 df-top 14177

This theorem is referenced by: basgen2 14260

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