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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 9828 | . . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} | |
2 | 0ss 4297 | . . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦 | |
3 | 2 | biantru 533 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) |
4 | ss0b 4298 | . . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅) | |
5 | 3, 4 | bitr3i 280 | . . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅) |
6 | 5 | anbi1ci 629 | . . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
7 | 6 | exbii 1855 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦))) |
8 | 0ex 5185 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
9 | fveq2 6695 | . . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅)) | |
10 | 9 | eqeq2d 2747 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅))) |
11 | 8, 10 | ceqsexv 3445 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅)) |
12 | card0 9539 | . . . . . . . . 9 ⊢ (card‘∅) = ∅ | |
13 | 12 | eqeq2i 2749 | . . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅) |
14 | 7, 11, 13 | 3bitri 300 | . . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅) |
15 | 14 | abbii 2801 | . . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅} |
16 | df-sn 4528 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
17 | 15, 16 | eqtr4i 2762 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅} |
18 | 17 | inteqi 4849 | . . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅} |
19 | 8 | intsn 4883 | . . . 4 ⊢ ∩ {∅} = ∅ |
20 | 18, 19 | eqtri 2759 | . . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅ |
21 | 1, 20 | sseqtri 3923 | . 2 ⊢ (cf‘∅) ⊆ ∅ |
22 | ss0b 4298 | . 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅) | |
23 | 21, 22 | mpbi 233 | 1 ⊢ (cf‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 {cab 2714 ⊆ wss 3853 ∅c0 4223 {csn 4527 ∪ cuni 4805 ∩ cint 4845 ‘cfv 6358 cardccrd 9516 cfccf 9518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-en 8605 df-card 9520 df-cf 9522 |
This theorem is referenced by: cfeq0 9835 cflim2 9842 cfidm 9854 alephsing 9855 alephreg 10161 pwcfsdom 10162 rankcf 10356 |
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