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Theorem nnsdomg 8764
 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 ssdomg 8541 . . 3 (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
2 ordom 7572 . . . 4 Ord ω
3 ordelss 6176 . . . 4 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
42, 3mpan 689 . . 3 (𝐴 ∈ ω → 𝐴 ⊆ ω)
51, 4impel 509 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
6 ominf 8717 . . . 4 ¬ ω ∈ Fin
7 ensym 8544 . . . . 5 (𝐴 ≈ ω → ω ≈ 𝐴)
8 breq2 5035 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
98rspcev 3571 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
10 isfi 8519 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
119, 10sylibr 237 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1211ex 416 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
137, 12syl5 34 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
146, 13mtoi 202 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
1514adantl 485 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
16 brsdom 8518 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
175, 15, 16sylanbrc 586 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   class class class wbr 5031  Ord word 6159  ωcom 7563   ≈ cen 8492   ≼ cdom 8493   ≺ csdm 8494  Fincfn 8495 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-om 7564  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499 This theorem is referenced by:  isfiniteg  8765  infsdomnn  8766  nnsdom  9104
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