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Theorem nnsdomg 9335
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 part of the antecedent. See nnsdom 9694 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5365. (Revised by BTernaryTau, 7-Jan-2025.)
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 7897 . . . . . 6 Ord ω
2 ordelss 6400 . . . . . 6 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 690 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
43adantr 480 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5 nnfi 9207 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ Fin)
6 ssdomfi2 9237 . . . . 5 ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
75, 6syl3an1 1164 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
84, 7mpd3an3 1464 . . 3 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
98ancoms 458 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10 ominf 9294 . . . 4 ¬ ω ∈ Fin
11 ensymfib 9224 . . . . . 6 (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
125, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13 breq2 5147 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
1413rspcev 3622 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15 isfi 9016 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
1614, 15sylibr 234 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1716ex 412 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
1812, 17sylbid 240 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
1910, 18mtoi 199 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
2019adantl 481 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21 brsdom 9015 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
229, 20, 21sylanbrc 583 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2108  wrex 3070  Vcvv 3480  wss 3951   class class class wbr 5143  Ord word 6383  ωcom 7887  cen 8982  cdom 8983  csdm 8984  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989
This theorem is referenced by:  isfiniteg  9337  infsdomnn  9338  infsdomnnOLD  9339  nnsdom  9694
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