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Theorem nnsdomg 8380
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 7235 . . . . 5 Ord ω
2 ordelss 5896 . . . . 5 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 708 . . . 4 (𝐴 ∈ ω → 𝐴 ⊆ ω)
4 ssdomg 8163 . . . 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 8333 . . . 4 ¬ ω ∈ Fin
8 ensym 8166 . . . . 5 (𝐴 ≈ ω → ω ≈ 𝐴)
9 breq2 4804 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
109rspcev 3445 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
11 isfi 8141 . . . . . . 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 473 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
17 brsdom 8140 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
186, 16, 17sylanbrc 701 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2135  wrex 3047  Vcvv 3336  wss 3711   class class class wbr 4800  Ord word 5879  ωcom 7226  cen 8114  cdom 8115  csdm 8116  Fincfn 8117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-om 7227  df-er 7907  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121
This theorem is referenced by:  isfiniteg  8381  infsdomnn  8382  nnsdom  8720
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