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Mirrors > Home > MPE Home > Th. List > cofss | Structured version Visualization version GIF version |
Description: Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.) |
Ref | Expression |
---|---|
cofss.1 | âĒ (ð â ðī â No ) |
cofss.2 | âĒ (ð â ðĩ â ðī) |
Ref | Expression |
---|---|
cofss | âĒ (ð â âðĨ â ðĩ âðĶ â ðī ðĨ âĪs ðĶ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofss.2 | . . . . 5 âĒ (ð â ðĩ â ðī) | |
2 | 1 | sselda 3980 | . . . 4 âĒ ((ð â§ ð§ â ðĩ) â ð§ â ðī) |
3 | cofss.1 | . . . . . . 7 âĒ (ð â ðī â No ) | |
4 | 1, 3 | sstrd 3990 | . . . . . 6 âĒ (ð â ðĩ â No ) |
5 | 4 | sselda 3980 | . . . . 5 âĒ ((ð â§ ð§ â ðĩ) â ð§ â No ) |
6 | slerflex 27695 | . . . . 5 âĒ (ð§ â No â ð§ âĪs ð§) | |
7 | 5, 6 | syl 17 | . . . 4 âĒ ((ð â§ ð§ â ðĩ) â ð§ âĪs ð§) |
8 | breq2 5152 | . . . . 5 âĒ (ðĶ = ð§ â (ð§ âĪs ðĶ â ð§ âĪs ð§)) | |
9 | 8 | rspcev 3609 | . . . 4 âĒ ((ð§ â ðī â§ ð§ âĪs ð§) â âðĶ â ðī ð§ âĪs ðĶ) |
10 | 2, 7, 9 | syl2anc 583 | . . 3 âĒ ((ð â§ ð§ â ðĩ) â âðĶ â ðī ð§ âĪs ðĶ) |
11 | 10 | ralrimiva 3143 | . 2 âĒ (ð â âð§ â ðĩ âðĶ â ðī ð§ âĪs ðĶ) |
12 | breq1 5151 | . . . 4 âĒ (ðĨ = ð§ â (ðĨ âĪs ðĶ â ð§ âĪs ðĶ)) | |
13 | 12 | rexbidv 3175 | . . 3 âĒ (ðĨ = ð§ â (âðĶ â ðī ðĨ âĪs ðĶ â âðĶ â ðī ð§ âĪs ðĶ)) |
14 | 13 | cbvralvw 3231 | . 2 âĒ (âðĨ â ðĩ âðĶ â ðī ðĨ âĪs ðĶ â âð§ â ðĩ âðĶ â ðī ð§ âĪs ðĶ) |
15 | 11, 14 | sylibr 233 | 1 âĒ (ð â âðĨ â ðĩ âðĶ â ðī ðĨ âĪs ðĶ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 395 â wcel 2099 âwral 3058 âwrex 3067 â wss 3947 class class class wbr 5148 No csur 27572 âĪs csle 27676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-1o 8486 df-2o 8487 df-no 27575 df-slt 27576 df-sle 27677 |
This theorem is referenced by: cutlt 27851 |
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