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Theorem domalom 34573
Description: A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.)
Assertion
Ref Expression
domalom (∀𝑛 ∈ ω 𝑛𝐴 → ¬ 𝐴 ∈ Fin)
Distinct variable group:   𝐴,𝑛

Proof of Theorem domalom
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfra1 3224 . . . 4 𝑛𝑛 ∈ ω 𝑛𝐴
2 breq1 5066 . . . . . . 7 (𝑦 = 𝑛 → (𝑦𝐴𝑛𝐴))
32imbi2d 342 . . . . . 6 (𝑦 = 𝑛 → ((∀𝑛 ∈ ω 𝑛𝐴𝑦𝐴) ↔ (∀𝑛 ∈ ω 𝑛𝐴𝑛𝐴)))
4 breq1 5066 . . . . . . 7 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ≺ 𝐴))
5 breq1 5066 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
6 breq1 5066 . . . . . . 7 (𝑦 = suc 𝑧 → (𝑦𝐴 ↔ suc 𝑧𝐴))
7 1n0 8115 . . . . . . . . 9 1o ≠ ∅
8 1onn 8260 . . . . . . . . . 10 1o ∈ ω
9 0sdomg 8640 . . . . . . . . . 10 (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
108, 9ax-mp 5 . . . . . . . . 9 (∅ ≺ 1o ↔ 1o ≠ ∅)
117, 10mpbir 232 . . . . . . . 8 ∅ ≺ 1o
12 breq1 5066 . . . . . . . . . 10 (𝑛 = 1o → (𝑛𝐴 ↔ 1o𝐴))
1312rspccv 3624 . . . . . . . . 9 (∀𝑛 ∈ ω 𝑛𝐴 → (1o ∈ ω → 1o𝐴))
148, 13mpi 20 . . . . . . . 8 (∀𝑛 ∈ ω 𝑛𝐴 → 1o𝐴)
15 sdomdomtr 8644 . . . . . . . 8 ((∅ ≺ 1o ∧ 1o𝐴) → ∅ ≺ 𝐴)
1611, 14, 15sylancr 587 . . . . . . 7 (∀𝑛 ∈ ω 𝑛𝐴 → ∅ ≺ 𝐴)
17 peano2 7595 . . . . . . . . . . 11 (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18 php4 8698 . . . . . . . . . . 11 (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
1917, 18syl 17 . . . . . . . . . 10 (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20 breq1 5066 . . . . . . . . . . . 12 (𝑛 = suc suc 𝑧 → (𝑛𝐴 ↔ suc suc 𝑧𝐴))
2120rspccv 3624 . . . . . . . . . . 11 (∀𝑛 ∈ ω 𝑛𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧𝐴))
22 peano2 7595 . . . . . . . . . . . 12 (suc 𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
2317, 22syl 17 . . . . . . . . . . 11 (𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
2421, 23impel 506 . . . . . . . . . 10 ((∀𝑛 ∈ ω 𝑛𝐴𝑧 ∈ ω) → suc suc 𝑧𝐴)
25 sdomdomtr 8644 . . . . . . . . . 10 ((suc 𝑧 ≺ suc suc 𝑧 ∧ suc suc 𝑧𝐴) → suc 𝑧𝐴)
2619, 24, 25syl2an2 682 . . . . . . . . 9 ((∀𝑛 ∈ ω 𝑛𝐴𝑧 ∈ ω) → suc 𝑧𝐴)
2726a1d 25 . . . . . . . 8 ((∀𝑛 ∈ ω 𝑛𝐴𝑧 ∈ ω) → (𝑧𝐴 → suc 𝑧𝐴))
2827expcom 414 . . . . . . 7 (𝑧 ∈ ω → (∀𝑛 ∈ ω 𝑛𝐴 → (𝑧𝐴 → suc 𝑧𝐴)))
294, 5, 6, 16, 28finds2 7603 . . . . . 6 (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛𝐴𝑦𝐴))
303, 29vtoclga 3579 . . . . 5 (𝑛 ∈ ω → (∀𝑛 ∈ ω 𝑛𝐴𝑛𝐴))
3130com12 32 . . . 4 (∀𝑛 ∈ ω 𝑛𝐴 → (𝑛 ∈ ω → 𝑛𝐴))
321, 31ralrimi 3221 . . 3 (∀𝑛 ∈ ω 𝑛𝐴 → ∀𝑛 ∈ ω 𝑛𝐴)
33 sdomnen 8532 . . . . 5 (𝑛𝐴 → ¬ 𝑛𝐴)
34 ensym 8552 . . . . 5 (𝐴𝑛𝑛𝐴)
3533, 34nsyl 142 . . . 4 (𝑛𝐴 → ¬ 𝐴𝑛)
3635ralimi 3165 . . 3 (∀𝑛 ∈ ω 𝑛𝐴 → ∀𝑛 ∈ ω ¬ 𝐴𝑛)
3732, 36syl 17 . 2 (∀𝑛 ∈ ω 𝑛𝐴 → ∀𝑛 ∈ ω ¬ 𝐴𝑛)
38 isfi 8527 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
3938notbii 321 . . 3 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴𝑛)
40 ralnex 3241 . . 3 (∀𝑛 ∈ ω ¬ 𝐴𝑛 ↔ ¬ ∃𝑛 ∈ ω 𝐴𝑛)
4139, 40bitr4i 279 . 2 𝐴 ∈ Fin ↔ ∀𝑛 ∈ ω ¬ 𝐴𝑛)
4237, 41sylibr 235 1 (∀𝑛 ∈ ω 𝑛𝐴 → ¬ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  c0 4295   class class class wbr 5063  suc csuc 6192  ωcom 7573  1oc1o 8091  cen 8500  cdom 8501  csdm 8502  Fincfn 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7574  df-1o 8098  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507
This theorem is referenced by:  isinf2  34574
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