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Mirrors > Home > MPE Home > Th. List > sslm | Structured version Visualization version Unicode version |
Description: A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.) |
Ref | Expression |
---|---|
sslm | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idd 24 | . . . . 5 | |
2 | idd 24 | . . . . 5 | |
3 | ssralv 3666 | . . . . 5 | |
4 | 1, 2, 3 | 3anim123d 1406 | . . . 4 |
5 | 4 | ssopab2dv 5004 | . . 3 |
6 | 5 | 3ad2ant3 1084 | . 2 TopOn TopOn |
7 | lmfval 21036 | . . 3 TopOn | |
8 | 7 | 3ad2ant2 1083 | . 2 TopOn TopOn |
9 | lmfval 21036 | . . 3 TopOn | |
10 | 9 | 3ad2ant1 1082 | . 2 TopOn TopOn |
11 | 6, 8, 10 | 3sstr4d 3648 | 1 TopOn TopOn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wss 3574 copab 4712 crn 5115 cres 5116 wf 5884 cfv 5888 (class class class)co 6650 cpm 7858 cc 9934 cuz 11687 TopOnctopon 20715 clm 21030 |
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-csb 3534 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-ov 6653 df-top 20699 df-topon 20716 df-lm 21033 |
This theorem is referenced by: (None) |
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