Theorem dcomex 9152

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 7462	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
2		df-br 4584	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3		elsni 4142	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4		fvex 6113	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6113	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 4870	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩ → (𝑓‘𝑛) = 1𝑜)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → (𝑓‘𝑛) = 1𝑜)
8	2, 7	sylbi 206	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1𝑜)
9		tz6.12i 6124	. . . . . . 7 ⊢ (1𝑜 ≠ ∅ → ((𝑓‘𝑛) = 1𝑜 → 𝑛𝑓1𝑜))
10	1, 8, 9	mpsyl 66	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11		vex 3176	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1on 7454	. . . . . . . 8 ⊢ 1𝑜 ∈ On
13	12	elexi 3186	. . . . . . 7 ⊢ 1𝑜 ∈ V
14	11, 13	breldm 5251	. . . . . 6 ⊢ (𝑛𝑓1𝑜 → 𝑛 ∈ dom 𝑓)
15	10, 14	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
16	15	ralimi 2936	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17		dfss3 3558	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
18	16, 17	sylibr 223	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
19		vex 3176	. . . . 5 ⊢ 𝑓 ∈ V
20	19	dmex 6991	. . . 4 ⊢ dom 𝑓 ∈ V
21	20	ssex 4730	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
22	18, 21	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
23		snex 4835	. . 3 ⊢ {⟨1𝑜, 1𝑜⟩} ∈ V
24	13, 13	fvsn 6351	. . . . . . . 8 ⊢ ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
25	13, 13	funsn 5853	. . . . . . . . 9 ⊢ Fun {⟨1𝑜, 1𝑜⟩}
26	13	snid 4155	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
27	13	dmsnop 5527	. . . . . . . . . 10 ⊢ dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
28	26, 27	eleqtrri 2687	. . . . . . . . 9 ⊢ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
29		funbrfvb 6148	. . . . . . . . 9 ⊢ ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
30	25, 28, 29	mp2an 704	. . . . . . . 8 ⊢ (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
31	24, 30	mpbi 219	. . . . . . 7 ⊢ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
32		breq12 4588	. . . . . . . 8 ⊢ ((𝑠 = 1𝑜 ∧ 𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
33	13, 13, 32	spc2ev 3274	. . . . . . 7 ⊢ (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
34	31, 33	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
35		breq 4585	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36	35	2exbidv 1839	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
37	34, 36	mpbiri 247	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
38		ssid 3587	. . . . . . 7 ⊢ {1𝑜} ⊆ {1𝑜}
39	13	rnsnop 5534	. . . . . . 7 ⊢ ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
40	38, 39, 27	3sstr4i 3607	. . . . . 6 ⊢ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
41		rneq 5272	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
42		dmeq 5246	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
43	41, 42	sseq12d 3597	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
44	40, 43	mpbiri 247	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 ⊆ dom 𝑥)
45		pm5.5 350	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46	37, 44, 45	syl2anc 691	. . . 4 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
47		breq 4585	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
48	47	ralbidv 2969	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
49	48	exbidv 1837	. . . 4 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
50	46, 49	bitrd 267	. . 3 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
51		ax-dc 9151	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
52	23, 50, 51	vtocl 3232	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
53	22, 52	exlimiiv 1846	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ran crn 5039  Oncon0 5640  suc csuc 5642  Fun wfun 5798  ‘cfv 5804  ωcom 6957  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-dc 9151

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1o 7447

This theorem is referenced by:  axdc2lem  9153  axdc3lem  9155  axdc4lem  9160  axcclem  9162