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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 10185 | . . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} | |
2 | 0ss 4356 | . . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦 | |
3 | 2 | biantru 530 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) |
4 | ss0b 4357 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅) | |
5 | 3, 4 | bitr3i 276 | . . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅) |
6 | 5 | anbi1ci 626 | . . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
7 | 6 | exbii 1850 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
8 | 0ex 5264 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
9 | fveq2 6842 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅)) | |
10 | 9 | eqeq2d 2747 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅))) |
11 | 8, 10 | ceqsexv 3494 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅)) |
12 | card0 9894 | . . . . . . . . 9 ⊢ (card‘∅) = ∅ | |
13 | 12 | eqeq2i 2749 | . . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅) |
14 | 7, 11, 13 | 3bitri 296 | . . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅) |
15 | 14 | abbii 2806 | . . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅} |
16 | df-sn 4587 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
17 | 15, 16 | eqtr4i 2767 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅} |
18 | 17 | inteqi 4911 | . . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅} |
19 | 8 | intsn 4947 | . . . 4 ⊢ ∩ {∅} = ∅ |
20 | 18, 19 | eqtri 2764 | . . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅ |
21 | 1, 20 | sseqtri 3980 | . 2 ⊢ (cf‘∅) ⊆ ∅ |
22 | ss0b 4357 | . 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅) | |
23 | 21, 22 | mpbi 229 | 1 ⊢ (cf‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 {cab 2713 ⊆ wss 3910 ∅c0 4282 {csn 4586 ∪ cuni 4865 ∩ cint 4907 ‘cfv 6496 cardccrd 9871 cfccf 9873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ord 6320 df-on 6321 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-en 8884 df-card 9875 df-cf 9877 |
This theorem is referenced by: cfeq0 10192 cflim2 10199 cfidm 10211 alephsing 10212 alephreg 10518 pwcfsdom 10519 rankcf 10713 |
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