Theorem cf0 10173

Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)

Assertion

Ref	Expression
cf0	⊢ (cf‘∅) = ∅

Proof of Theorem cf0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfub 10171	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 4354	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 529	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 4355	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 277	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi1ci 627	. . . . . . . . 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 5254	. . . . . . . . 9 ⊢ ∅ ∈ V
9		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
10	9	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
11	8, 10	ceqsexv 3492	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12		card0 9882	. . . . . . . . 9 ⊢ (card‘∅) = ∅
13	12	eqeq2i 2750	. . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
14	7, 11, 13	3bitri 297	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
15	14	abbii 2804	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
16		df-sn 4583	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
17	15, 16	eqtr4i 2763	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
18	17	inteqi 4908	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
19	8	intsn 4941	. . . 4 ⊢ ∩ {∅} = ∅
20	18, 19	eqtri 2760	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
21	1, 20	sseqtri 3984	. 2 ⊢ (cf‘∅) ⊆ ∅
22		ss0b 4355	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
23	21, 22	mpbi 230	1 ⊢ (cf‘∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865  ∩ cint 4904  ‘cfv 6500  cardccrd 9859  cfccf 9861

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8896  df-card 9863  df-cf 9865

This theorem is referenced by:  cfeq0  10178  cflim2  10185  cfidm  10197  alephsing  10198  alephreg  10505  pwcfsdom  10506  rankcf  10700