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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 3975 | . . . 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 3985 | . . . . . 6 âĒ (ð â ðĩ â No ) |
5 | 4 | sselda 3975 | . . . . 5 âĒ ((ð â§ ð§ â ðĩ) â ð§ â No ) |
6 | slerflex 27636 | . . . . 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 5143 | . . . . 5 âĒ (ðĶ = ð§ â (ð§ âĪs ðĶ â ð§ âĪs ð§)) | |
9 | 8 | rspcev 3604 | . . . 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 3138 | . 2 âĒ (ð â âð§ â ðĩ âðĶ â ðī ð§ âĪs ðĶ) |
12 | breq1 5142 | . . . 4 âĒ (ðĨ = ð§ â (ðĨ âĪs ðĶ â ð§ âĪs ðĶ)) | |
13 | 12 | rexbidv 3170 | . . 3 âĒ (ðĨ = ð§ â (âðĶ â ðī ðĨ âĪs ðĶ â âðĶ â ðī ð§ âĪs ðĶ)) |
14 | 13 | cbvralvw 3226 | . 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 2098 âwral 3053 âwrex 3062 â wss 3941 class class class wbr 5139 No csur 27513 âĪs csle 27617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-1o 8462 df-2o 8463 df-no 27516 df-slt 27517 df-sle 27618 |
This theorem is referenced by: cutlt 27792 |
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