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Theorem nnsdomg 8077
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 6939 . . . . 5 Ord ω
2 ordelss 5638 . . . . 5 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 701 . . . 4 (𝐴 ∈ ω → 𝐴 ⊆ ω)
4 ssdomg 7860 . . . 4 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
53, 4syl5 33 . . 3 (ω ∈ V → (𝐴 ∈ ω → 𝐴 ≼ ω))
65imp 443 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
7 ominf 8030 . . . 4 ¬ ω ∈ Fin
8 ensym 7864 . . . . 5 (𝐴 ≈ ω → ω ≈ 𝐴)
9 breq2 4577 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
109rspcev 3277 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
11 isfi 7838 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
1210, 11sylibr 222 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1312ex 448 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
148, 13syl5 33 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
157, 14mtoi 188 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
1615adantl 480 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
17 brsdom 7837 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
186, 16, 17sylanbrc 694 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 1975  wrex 2892  Vcvv 3168  wss 3535   class class class wbr 4573  Ord word 5621  ωcom 6930  cen 7811  cdom 7812  csdm 7813  Fincfn 7814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818
This theorem is referenced by:  isfiniteg  8078  infsdomnn  8079  nnsdom  8407
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