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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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