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Theorem basgen 11947

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 11930	. . . 4 $/\$
2	1	3adant3 966	. . 3 $/\$ $/\$
3		tgtop 11935	. . . 4
4	3	3ad2ant1 967	. . 3 $/\$ $/\$
5	2, 4	sseqtrd 3077	. 2 $/\$ $/\$
6		simp3 948	. 2 $/\$ $/\$
7	5, 6	eqssd 3056	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ w3a 927

wceq 1296

wcel 1445

wss 3013

cfv 5049

ctg 11834

ctop 11863

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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284

This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-topgen 11840 df-top 11864

This theorem is referenced by: basgen2 11948

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