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Theorem dcomex 9220
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 7527 . . . . . . 7 1𝑜 ≠ ∅
2 df-br 4619 . . . . . . . 8 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3 elsni 4170 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4 fvex 6163 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6163 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 4909 . . . . . . . . 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 6176 . . . . . . 7 (1𝑜 ≠ ∅ → ((𝑓𝑛) = 1𝑜𝑛𝑓1𝑜))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11 vex 3192 . . . . . . 7 𝑛 ∈ V
12 1on 7519 . . . . . . . 8 1𝑜 ∈ On
1312elexi 3202 . . . . . . 7 1𝑜 ∈ V
1411, 13breldm 5294 . . . . . 6 (𝑛𝑓1𝑜𝑛 ∈ dom 𝑓)
1510, 14syl 17 . . . . 5 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1615ralimi 2947 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17 dfss3 3577 . . . 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 3192 . . . . 5 𝑓 ∈ V
2019dmex 7053 . . . 4 dom 𝑓 ∈ V
2120ssex 4767 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2218, 21syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
23 snex 4874 . . 3 {⟨1𝑜, 1𝑜⟩} ∈ V
2413, 13fvsn 6406 . . . . . . . 8 ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
2513, 13funsn 5902 . . . . . . . . 9 Fun {⟨1𝑜, 1𝑜⟩}
2613snid 4184 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
2713dmsnop 5573 . . . . . . . . . 10 dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
2826, 27eleqtrri 2697 . . . . . . . . 9 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
29 funbrfvb 6200 . . . . . . . . 9 ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3025, 28, 29mp2an 707 . . . . . . . 8 (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
3124, 30mpbi 220 . . . . . . 7 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
32 breq12 4623 . . . . . . . 8 ((𝑠 = 1𝑜𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3313, 13, 32spc2ev 3290 . . . . . . 7 (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
3431, 33ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
35 breq 4620 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36352exbidv 1849 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
3734, 36mpbiri 248 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
38 ssid 3608 . . . . . . 7 {1𝑜} ⊆ {1𝑜}
3913rnsnop 5580 . . . . . . 7 ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
4038, 39, 273sstr4i 3628 . . . . . 6 ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
41 rneq 5316 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
42 dmeq 5289 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
4341, 42sseq12d 3618 . . . . . 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 pm5.5 351 . . . . 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 692 . . . 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 4620 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4847ralbidv 2981 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4948exbidv 1847 . . . 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 9219 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5223, 50, 51vtocl 3248 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
5322, 52exlimiiv 1856 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3189  wss 3559  c0 3896  {csn 4153  cop 4159   class class class wbr 4618  dom cdm 5079  ran crn 5080  Oncon0 5687  suc csuc 5689  Fun wfun 5846  cfv 5852  ωcom 7019  1𝑜c1o 7505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909  ax-dc 9219
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-1o 7512
This theorem is referenced by:  axdc2lem  9221  axdc3lem  9223  axdc4lem  9228  axcclem  9230
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