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| 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 10290 | . . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} | |
| 2 | 0ss 4399 | . . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦 | |
| 3 | 2 | biantru 529 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) | 
| 4 | ss0b 4400 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅) | |
| 5 | 3, 4 | bitr3i 277 | . . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅) | 
| 6 | 5 | anbi1ci 626 | . . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) | 
| 7 | 6 | exbii 1847 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) | 
| 8 | 0ex 5306 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
| 9 | fveq2 6905 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅)) | |
| 10 | 9 | eqeq2d 2747 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅))) | 
| 11 | 8, 10 | ceqsexv 3531 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅)) | 
| 12 | card0 9999 | . . . . . . . . 9 ⊢ (card‘∅) = ∅ | |
| 13 | 12 | eqeq2i 2749 | . . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅) | 
| 14 | 7, 11, 13 | 3bitri 297 | . . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅) | 
| 15 | 14 | abbii 2808 | . . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅} | 
| 16 | df-sn 4626 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
| 17 | 15, 16 | eqtr4i 2767 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅} | 
| 18 | 17 | inteqi 4949 | . . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅} | 
| 19 | 8 | intsn 4983 | . . . 4 ⊢ ∩ {∅} = ∅ | 
| 20 | 18, 19 | eqtri 2764 | . . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅ | 
| 21 | 1, 20 | sseqtri 4031 | . 2 ⊢ (cf‘∅) ⊆ ∅ | 
| 22 | ss0b 4400 | . 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅) | |
| 23 | 21, 22 | mpbi 230 | 1 ⊢ (cf‘∅) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∃wex 1778 {cab 2713 ⊆ wss 3950 ∅c0 4332 {csn 4625 ∪ cuni 4906 ∩ cint 4945 ‘cfv 6560 cardccrd 9976 cfccf 9978 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-en 8987 df-card 9980 df-cf 9982 | 
| This theorem is referenced by: cfeq0 10297 cflim2 10304 cfidm 10316 alephsing 10317 alephreg 10623 pwcfsdom 10624 rankcf 10818 | 
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