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Mirrors > Home > MPE Home > Th. List > cf0 | Structured version Visualization version GIF version |
Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.) |
Ref | Expression |
---|---|
cf0 | ⊢ (cf‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfub 9660 | . . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} | |
2 | 0ss 4349 | . . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦 | |
3 | 2 | biantru 530 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) |
4 | ss0b 4350 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅) | |
5 | 3, 4 | bitr3i 278 | . . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅) |
6 | 5 | anbi1ci 625 | . . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
7 | 6 | exbii 1839 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
8 | 0ex 5203 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
9 | fveq2 6664 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅)) | |
10 | 9 | eqeq2d 2832 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅))) |
11 | 8, 10 | ceqsexv 3542 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅)) |
12 | card0 9376 | . . . . . . . . 9 ⊢ (card‘∅) = ∅ | |
13 | 12 | eqeq2i 2834 | . . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅) |
14 | 7, 11, 13 | 3bitri 298 | . . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅) |
15 | 14 | abbii 2886 | . . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅} |
16 | df-sn 4560 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
17 | 15, 16 | eqtr4i 2847 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅} |
18 | 17 | inteqi 4873 | . . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅} |
19 | 8 | intsn 4905 | . . . 4 ⊢ ∩ {∅} = ∅ |
20 | 18, 19 | eqtri 2844 | . . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅ |
21 | 1, 20 | sseqtri 4002 | . 2 ⊢ (cf‘∅) ⊆ ∅ |
22 | ss0b 4350 | . 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅) | |
23 | 21, 22 | mpbi 231 | 1 ⊢ (cf‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 {cab 2799 ⊆ wss 3935 ∅c0 4290 {csn 4559 ∪ cuni 4832 ∩ cint 4869 ‘cfv 6349 cardccrd 9353 cfccf 9355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-en 8499 df-card 9357 df-cf 9359 |
This theorem is referenced by: cfeq0 9667 cflim2 9674 cfidm 9686 alephsing 9687 alephreg 9993 pwcfsdom 9994 rankcf 10188 |
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