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Theorem isfinite2 8103
 Description: Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
isfinite2 (𝐴 ≺ ω → 𝐴 ∈ Fin)

Proof of Theorem isfinite2
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsdom 7848 . . 3 Rel ≺
21brrelex2i 5083 . 2 (𝐴 ≺ ω → ω ∈ V)
3 sdomdom 7869 . . . 4 (𝐴 ≺ ω → 𝐴 ≼ ω)
4 domeng 7855 . . . 4 (ω ∈ V → (𝐴 ≼ ω ↔ ∃𝑦(𝐴𝑦𝑦 ⊆ ω)))
53, 4syl5ib 233 . . 3 (ω ∈ V → (𝐴 ≺ ω → ∃𝑦(𝐴𝑦𝑦 ⊆ ω)))
6 ensym 7891 . . . . . . . . . . 11 (𝐴𝑦𝑦𝐴)
76ad2antrl 760 . . . . . . . . . 10 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝑦𝐴)
8 simpl 472 . . . . . . . . . 10 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝐴 ≺ ω)
9 ensdomtr 7981 . . . . . . . . . 10 ((𝑦𝐴𝐴 ≺ ω) → 𝑦 ≺ ω)
107, 8, 9syl2anc 691 . . . . . . . . 9 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝑦 ≺ ω)
11 sdomnen 7870 . . . . . . . . 9 (𝑦 ≺ ω → ¬ 𝑦 ≈ ω)
1210, 11syl 17 . . . . . . . 8 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → ¬ 𝑦 ≈ ω)
13 simpr 476 . . . . . . . . 9 ((𝐴𝑦𝑦 ⊆ ω) → 𝑦 ⊆ ω)
14 unbnn 8101 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑦 ⊆ ω ∧ ∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤) → 𝑦 ≈ ω)
15143expia 1259 . . . . . . . . 9 ((ω ∈ V ∧ 𝑦 ⊆ ω) → (∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤𝑦 ≈ ω))
162, 13, 15syl2an 493 . . . . . . . 8 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → (∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤𝑦 ≈ ω))
1712, 16mtod 188 . . . . . . 7 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → ¬ ∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤)
18 rexnal 2978 . . . . . . . . 9 (∃𝑧 ∈ ω ¬ ∃𝑤𝑦 𝑧𝑤 ↔ ¬ ∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤)
19 omsson 6961 . . . . . . . . . . . . 13 ω ⊆ On
20 sstr 3576 . . . . . . . . . . . . 13 ((𝑦 ⊆ ω ∧ ω ⊆ On) → 𝑦 ⊆ On)
2119, 20mpan2 703 . . . . . . . . . . . 12 (𝑦 ⊆ ω → 𝑦 ⊆ On)
22 nnord 6965 . . . . . . . . . . . 12 (𝑧 ∈ ω → Ord 𝑧)
23 ssel2 3563 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑤𝑦) → 𝑤 ∈ On)
24 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
2524elon 5649 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ On ↔ Ord 𝑤)
2623, 25sylib 207 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ On ∧ 𝑤𝑦) → Ord 𝑤)
27 ordtri1 5673 . . . . . . . . . . . . . . . . 17 ((Ord 𝑤 ∧ Ord 𝑧) → (𝑤𝑧 ↔ ¬ 𝑧𝑤))
2826, 27sylan 487 . . . . . . . . . . . . . . . 16 (((𝑦 ⊆ On ∧ 𝑤𝑦) ∧ Ord 𝑧) → (𝑤𝑧 ↔ ¬ 𝑧𝑤))
2928an32s 842 . . . . . . . . . . . . . . 15 (((𝑦 ⊆ On ∧ Ord 𝑧) ∧ 𝑤𝑦) → (𝑤𝑧 ↔ ¬ 𝑧𝑤))
3029ralbidva 2968 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ Ord 𝑧) → (∀𝑤𝑦 𝑤𝑧 ↔ ∀𝑤𝑦 ¬ 𝑧𝑤))
31 unissb 4405 . . . . . . . . . . . . . 14 ( 𝑦𝑧 ↔ ∀𝑤𝑦 𝑤𝑧)
32 ralnex 2975 . . . . . . . . . . . . . . 15 (∀𝑤𝑦 ¬ 𝑧𝑤 ↔ ¬ ∃𝑤𝑦 𝑧𝑤)
3332bicomi 213 . . . . . . . . . . . . . 14 (¬ ∃𝑤𝑦 𝑧𝑤 ↔ ∀𝑤𝑦 ¬ 𝑧𝑤)
3430, 31, 333bitr4g 302 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ Ord 𝑧) → ( 𝑦𝑧 ↔ ¬ ∃𝑤𝑦 𝑧𝑤))
35 ordunisssuc 5747 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ Ord 𝑧) → ( 𝑦𝑧𝑦 ⊆ suc 𝑧))
3634, 35bitr3d 269 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ Ord 𝑧) → (¬ ∃𝑤𝑦 𝑧𝑤𝑦 ⊆ suc 𝑧))
3721, 22, 36syl2an 493 . . . . . . . . . . 11 ((𝑦 ⊆ ω ∧ 𝑧 ∈ ω) → (¬ ∃𝑤𝑦 𝑧𝑤𝑦 ⊆ suc 𝑧))
38 peano2b 6973 . . . . . . . . . . . . . 14 (𝑧 ∈ ω ↔ suc 𝑧 ∈ ω)
39 ssnnfi 8064 . . . . . . . . . . . . . 14 ((suc 𝑧 ∈ ω ∧ 𝑦 ⊆ suc 𝑧) → 𝑦 ∈ Fin)
4038, 39sylanb 488 . . . . . . . . . . . . 13 ((𝑧 ∈ ω ∧ 𝑦 ⊆ suc 𝑧) → 𝑦 ∈ Fin)
4140ex 449 . . . . . . . . . . . 12 (𝑧 ∈ ω → (𝑦 ⊆ suc 𝑧𝑦 ∈ Fin))
4241adantl 481 . . . . . . . . . . 11 ((𝑦 ⊆ ω ∧ 𝑧 ∈ ω) → (𝑦 ⊆ suc 𝑧𝑦 ∈ Fin))
4337, 42sylbid 229 . . . . . . . . . 10 ((𝑦 ⊆ ω ∧ 𝑧 ∈ ω) → (¬ ∃𝑤𝑦 𝑧𝑤𝑦 ∈ Fin))
4443rexlimdva 3013 . . . . . . . . 9 (𝑦 ⊆ ω → (∃𝑧 ∈ ω ¬ ∃𝑤𝑦 𝑧𝑤𝑦 ∈ Fin))
4518, 44syl5bir 232 . . . . . . . 8 (𝑦 ⊆ ω → (¬ ∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤𝑦 ∈ Fin))
4645ad2antll 761 . . . . . . 7 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → (¬ ∀𝑧 ∈ ω ∃𝑤𝑦 𝑧𝑤𝑦 ∈ Fin))
4717, 46mpd 15 . . . . . 6 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝑦 ∈ Fin)
48 simprl 790 . . . . . 6 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝐴𝑦)
49 enfii 8062 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐴𝑦) → 𝐴 ∈ Fin)
5047, 48, 49syl2anc 691 . . . . 5 ((𝐴 ≺ ω ∧ (𝐴𝑦𝑦 ⊆ ω)) → 𝐴 ∈ Fin)
5150ex 449 . . . 4 (𝐴 ≺ ω → ((𝐴𝑦𝑦 ⊆ ω) → 𝐴 ∈ Fin))
5251exlimdv 1848 . . 3 (𝐴 ≺ ω → (∃𝑦(𝐴𝑦𝑦 ⊆ ω) → 𝐴 ∈ Fin))
535, 52sylcom 30 . 2 (ω ∈ V → (𝐴 ≺ ω → 𝐴 ∈ Fin))
542, 53mpcom 37 1 (𝐴 ≺ ω → 𝐴 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Ord word 5639  Oncon0 5640  suc csuc 5642  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841 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-pow 4769  ax-pr 4833  ax-un 6847 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-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  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-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845 This theorem is referenced by:  isfiniteg  8105  unfi2  8114  unifi2  8139  axcclem  9162  dirith2  25017  padct  28885  volmeas  29621  axccdom  38411  axccd2  38425
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