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Mirrors > Home > MPE Home > Th. List > t1t0 | Structured version Visualization version Unicode version |
Description: A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
t1t0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1top 21134 | . . 3 | |
2 | eqid 2622 | . . . 4 | |
3 | 2 | toptopon 20722 | . . 3 TopOn |
4 | 1, 3 | sylib 208 | . 2 TopOn |
5 | biimp 205 | . . . . . . . 8 | |
6 | 5 | ralimi 2952 | . . . . . . 7 |
7 | 6 | imim1i 63 | . . . . . 6 |
8 | 7 | ralimi 2952 | . . . . 5 |
9 | 8 | ralimi 2952 | . . . 4 |
10 | 9 | a1i 11 | . . 3 TopOn |
11 | ist1-2 21151 | . . 3 TopOn | |
12 | ist0-2 21148 | . . 3 TopOn | |
13 | 10, 11, 12 | 3imtr4d 283 | . 2 TopOn |
14 | 4, 13 | mpcom 38 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wcel 1990 wral 2912 cuni 4436 cfv 5888 ctop 20698 TopOnctopon 20715 ct0 21110 ct1 21111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 df-top 20699 df-topon 20716 df-cld 20823 df-t0 21117 df-t1 21118 |
This theorem is referenced by: t1r0 21624 ist1-5 21625 ishaus3 21626 reghaus 21628 nrmhaus 21629 tgpt0 21922 |
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