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Theorem basgen 14803

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 14786	. . . 4 $/\$
2	1	3adant3 1043	. . 3 $/\$ $/\$
3		tgtop 14791	. . . 4
4	3	3ad2ant1 1044	. . 3 $/\$ $/\$
5	2, 4	sseqtrd 3265	. 2 $/\$ $/\$
6		simp3 1025	. 2 $/\$ $/\$
7	5, 6	eqssd 3244	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ w3a 1004

wceq 1397

wcel 2202

wss 3200

cfv 5326

ctg 13336

ctop 14720

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-sbc 3032 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-topgen 13342 df-top 14721

This theorem is referenced by: basgen2 14804

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