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Theorem basgen 14794

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 14777	. . . 4 $/\$
2	1	3adant3 1041	. . 3 $/\$ $/\$
3		tgtop 14782	. . . 4
4	3	3ad2ant1 1042	. . 3 $/\$ $/\$
5	2, 4	sseqtrd 3263	. 2 $/\$ $/\$
6		simp3 1023	. 2 $/\$ $/\$
7	5, 6	eqssd 3242	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ w3a 1002

wceq 1395

wcel 2200

wss 3198

cfv 5324

ctg 13327

ctop 14711

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2802 df-sbc 3030 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-topgen 13333 df-top 14712

This theorem is referenced by: basgen2 14795

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