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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.
StepHypRef Expression
1 1n0 7462 . . . . . . 7 1𝑜 ≠ ∅
2 df-br 4584 . . . . . . . 8 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3 elsni 4142 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4 fvex 6113 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6113 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 4870 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩ → (𝑓𝑛) = 1𝑜)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → (𝑓𝑛) = 1𝑜)
82, 7sylbi 206 . . . . . . 7 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1𝑜)
9 tz6.12i 6124 . . . . . . 7 (1𝑜 ≠ ∅ → ((𝑓𝑛) = 1𝑜𝑛𝑓1𝑜))
101, 8, 9mpsyl 66 . . . . . 6 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11 vex 3176 . . . . . . 7 𝑛 ∈ V
12 1on 7454 . . . . . . . 8 1𝑜 ∈ On
1312elexi 3186 . . . . . . 7 1𝑜 ∈ V
1411, 13breldm 5251 . . . . . 6 (𝑛𝑓1𝑜𝑛 ∈ dom 𝑓)
1510, 14syl 17 . . . . 5 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1615ralimi 2936 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17 dfss3 3558 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1816, 17sylibr 223 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
19 vex 3176 . . . . 5 𝑓 ∈ V
2019dmex 6991 . . . 4 dom 𝑓 ∈ V
2120ssex 4730 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2218, 21syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
23 snex 4835 . . 3 {⟨1𝑜, 1𝑜⟩} ∈ V
2413, 13fvsn 6351 . . . . . . . 8 ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
2513, 13funsn 5853 . . . . . . . . 9 Fun {⟨1𝑜, 1𝑜⟩}
2613snid 4155 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
2713dmsnop 5527 . . . . . . . . . 10 dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
2826, 27eleqtrri 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))
3025, 28, 29mp2an 704 . . . . . . . 8 (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
3124, 30mpbi 219 . . . . . . 7 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
32 breq12 4588 . . . . . . . 8 ((𝑠 = 1𝑜𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3313, 13, 32spc2ev 3274 . . . . . . 7 (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
3431, 33ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
35 breq 4585 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36352exbidv 1839 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
3734, 36mpbiri 247 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
38 ssid 3587 . . . . . . 7 {1𝑜} ⊆ {1𝑜}
3913rnsnop 5534 . . . . . . 7 ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
4038, 39, 273sstr4i 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⟩})
4341, 42sseq12d 3597 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
4440, 43mpbiri 247 . . . . 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 691 . . . 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 4585 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4847ralbidv 2969 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4948exbidv 1837 . . . 4 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
5046, 49bitrd 267 . . 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 9151 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5223, 50, 51vtocl 3232 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
5322, 52exlimiiv 1846 1 ω ∈ V
 Colors of variables: wff setvar class 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  1𝑜c1o 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
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