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Theorem nnsdomg 8260
 Description: Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
Assertion
Ref Expression
nnsdomg ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Proof of Theorem nnsdomg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordom 7116 . . . . 5 Ord ω
2 ordelss 5777 . . . . 5 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 706 . . . 4 (𝐴 ∈ ω → 𝐴 ⊆ ω)
4 ssdomg 8043 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
53, 4syl5 34 . . 3 (ω ∈ V → (𝐴 ∈ ω → 𝐴 ≼ ω))
65imp 444 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
7 ominf 8213 . . . 4 ¬ ω ∈ Fin
8 ensym 8046 . . . . 5 (𝐴 ≈ ω → ω ≈ 𝐴)
9 breq2 4689 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
109rspcev 3340 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
11 isfi 8021 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
1210, 11sylibr 224 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1312ex 449 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
148, 13syl5 34 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
157, 14mtoi 190 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
1615adantl 481 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
17 brsdom 8020 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
186, 16, 17sylanbrc 699 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685  Ord word 5760  ωcom 7107   ≈ cen 7994   ≼ cdom 7995   ≺ csdm 7996  Fincfn 7997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001 This theorem is referenced by:  isfiniteg  8261  infsdomnn  8262  nnsdom  8589
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