![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9109 | . . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} | |
2 | 0ss 4005 | . . . . . . . . . . . . 13 ⊢ ∅ ⊆ ∪ 𝑦 | |
3 | 2 | biantru 525 | . . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) |
4 | ss0b 4006 | . . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅) | |
5 | 3, 4 | bitr3i 266 | . . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅) |
6 | 5 | anbi2i 730 | . . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑥 = (card‘𝑦) ∧ 𝑦 = ∅)) |
7 | ancom 465 | . . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ 𝑦 = ∅) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) | |
8 | 6, 7 | bitri 264 | . . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
9 | 8 | exbii 1814 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
10 | 0ex 4823 | . . . . . . . . . 10 ⊢ ∅ ∈ V | |
11 | fveq2 6229 | . . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅)) | |
12 | 11 | eqeq2d 2661 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅))) |
13 | 10, 12 | ceqsexv 3273 | . . . . . . . . 9 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅)) |
14 | card0 8822 | . . . . . . . . . 10 ⊢ (card‘∅) = ∅ | |
15 | 14 | eqeq2i 2663 | . . . . . . . . 9 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅) |
16 | 13, 15 | bitri 264 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = ∅) |
17 | 9, 16 | bitri 264 | . . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅) |
18 | 17 | abbii 2768 | . . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅} |
19 | df-sn 4211 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
20 | 18, 19 | eqtr4i 2676 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅} |
21 | 20 | inteqi 4511 | . . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅} |
22 | 10 | intsn 4545 | . . . 4 ⊢ ∩ {∅} = ∅ |
23 | 21, 22 | eqtri 2673 | . . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅ |
24 | 1, 23 | sseqtri 3670 | . 2 ⊢ (cf‘∅) ⊆ ∅ |
25 | ss0b 4006 | . 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅) | |
26 | 24, 25 | mpbi 220 | 1 ⊢ (cf‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∃wex 1744 {cab 2637 ⊆ wss 3607 ∅c0 3948 {csn 4210 ∪ cuni 4468 ∩ cint 4507 ‘cfv 5926 cardccrd 8799 cfccf 8801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-en 7998 df-card 8803 df-cf 8805 |
This theorem is referenced by: cfeq0 9116 cflim2 9123 cfidm 9135 alephsing 9136 alephreg 9442 pwcfsdom 9443 rankcf 9637 |
Copyright terms: Public domain | W3C validator |