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Theorem dcomex 9307
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.
StepHypRef Expression
1 1n0 7620 . . . . . . 7 1𝑜 ≠ ∅
2 df-br 4686 . . . . . . . 8 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3 elsni 4227 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4 fvex 6239 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6239 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 4973 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩ → (𝑓𝑛) = 1𝑜)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → (𝑓𝑛) = 1𝑜)
82, 7sylbi 207 . . . . . . 7 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1𝑜)
9 tz6.12i 6252 . . . . . . 7 (1𝑜 ≠ ∅ → ((𝑓𝑛) = 1𝑜𝑛𝑓1𝑜))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11 vex 3234 . . . . . . 7 𝑛 ∈ V
12 1on 7612 . . . . . . . 8 1𝑜 ∈ On
1312elexi 3244 . . . . . . 7 1𝑜 ∈ V
1411, 13breldm 5361 . . . . . 6 (𝑛𝑓1𝑜𝑛 ∈ dom 𝑓)
1510, 14syl 17 . . . . 5 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1615ralimi 2981 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17 dfss3 3625 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1816, 17sylibr 224 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
19 vex 3234 . . . . 5 𝑓 ∈ V
2019dmex 7141 . . . 4 dom 𝑓 ∈ V
2120ssex 4835 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2218, 21syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
23 snex 4938 . . 3 {⟨1𝑜, 1𝑜⟩} ∈ V
2413, 13fvsn 6487 . . . . . . . 8 ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
2513, 13funsn 5977 . . . . . . . . 9 Fun {⟨1𝑜, 1𝑜⟩}
2613snid 4241 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
2713dmsnop 5645 . . . . . . . . . 10 dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
2826, 27eleqtrri 2729 . . . . . . . . 9 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
29 funbrfvb 6276 . . . . . . . . 9 ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3025, 28, 29mp2an 708 . . . . . . . 8 (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
3124, 30mpbi 220 . . . . . . 7 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
32 breq12 4690 . . . . . . . 8 ((𝑠 = 1𝑜𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3313, 13, 32spc2ev 3332 . . . . . . 7 (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
3431, 33ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
35 breq 4687 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36352exbidv 1892 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
3734, 36mpbiri 248 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
38 ssid 3657 . . . . . . 7 {1𝑜} ⊆ {1𝑜}
3913rnsnop 5653 . . . . . . 7 ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
4038, 39, 273sstr4i 3677 . . . . . 6 ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
41 rneq 5383 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
42 dmeq 5356 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
4341, 42sseq12d 3667 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
4440, 43mpbiri 248 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 ⊆ dom 𝑥)
45 pm5.5 350 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4637, 44, 45syl2anc 694 . . . 4 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
47 breq 4687 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4847ralbidv 3015 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4948exbidv 1890 . . . 4 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
5046, 49bitrd 268 . . 3 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
51 ax-dc 9306 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5223, 50, 51vtocl 3290 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
5322, 52exlimiiv 1899 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  c0 3948  {csn 4210  cop 4216   class class class wbr 4685  dom cdm 5143  ran crn 5144  Oncon0 5761  suc csuc 5763  Fun wfun 5920  cfv 5926  ωcom 7107  1𝑜c1o 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-dc 9306
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-1o 7605
This theorem is referenced by:  axdc2lem  9308  axdc3lem  9310  axdc4lem  9315  axcclem  9317
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