Theorem dcomex 9269

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 7575	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
2		df-br 4654	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3		elsni 4194	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4		fvex 6201	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6201	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 4944	. . . . . . . . 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 207	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1𝑜)
9		tz6.12i 6214	. . . . . . 7 ⊢ (1𝑜 ≠ ∅ → ((𝑓‘𝑛) = 1𝑜 → 𝑛𝑓1𝑜))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11		vex 3203	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1on 7567	. . . . . . . 8 ⊢ 1𝑜 ∈ On
13	12	elexi 3213	. . . . . . 7 ⊢ 1𝑜 ∈ V
14	11, 13	breldm 5329	. . . . . 6 ⊢ (𝑛𝑓1𝑜 → 𝑛 ∈ dom 𝑓)
15	10, 14	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
16	15	ralimi 2952	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17		dfss3 3592	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
18	16, 17	sylibr 224	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
19		vex 3203	. . . . 5 ⊢ 𝑓 ∈ V
20	19	dmex 7099	. . . 4 ⊢ dom 𝑓 ∈ V
21	20	ssex 4802	. . 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 4908	. . 3 ⊢ {⟨1𝑜, 1𝑜⟩} ∈ V
24	13, 13	fvsn 6446	. . . . . . . 8 ⊢ ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
25	13, 13	funsn 5939	. . . . . . . . 9 ⊢ Fun {⟨1𝑜, 1𝑜⟩}
26	13	snid 4208	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
27	13	dmsnop 5609	. . . . . . . . . 10 ⊢ dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
28	26, 27	eleqtrri 2700	. . . . . . . . 9 ⊢ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
29		funbrfvb 6238	. . . . . . . . 9 ⊢ ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
30	25, 28, 29	mp2an 708	. . . . . . . 8 ⊢ (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
31	24, 30	mpbi 220	. . . . . . 7 ⊢ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
32		breq12 4658	. . . . . . . 8 ⊢ ((𝑠 = 1𝑜 ∧ 𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
33	13, 13, 32	spc2ev 3301	. . . . . . 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 4655	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36	35	2exbidv 1852	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
37	34, 36	mpbiri 248	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
38		ssid 3624	. . . . . . 7 ⊢ {1𝑜} ⊆ {1𝑜}
39	13	rnsnop 5616	. . . . . . 7 ⊢ ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
40	38, 39, 27	3sstr4i 3644	. . . . . 6 ⊢ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
41		rneq 5351	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
42		dmeq 5324	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
43	41, 42	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
44	40, 43	mpbiri 248	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 ⊆ dom 𝑥)
45		pm5.5 351	. . . . 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 693	. . . 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 4655	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
48	47	ralbidv 2986	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
49	48	exbidv 1850	. . . 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 268	. . 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 9268	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
52	23, 50, 51	vtocl 3259	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
53	22, 52	exlimiiv 1859	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  ran crn 5115  Oncon0 5723  suc csuc 5725  Fun wfun 5882  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-dc 9268

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-1o 7560

This theorem is referenced by:  axdc2lem  9270  axdc3lem  9272  axdc4lem  9277  axcclem  9279