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Theorem nnsdomg 9253
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 9597 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5325. (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 7817 . . . . . 6 Ord ω
2 ordelss 6338 . . . . . 6 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 689 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
43adantr 482 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5 nnfi 9118 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ Fin)
6 ssdomfi2 9151 . . . . 5 ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
75, 6syl3an1 1164 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
84, 7mpd3an3 1463 . . 3 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
98ancoms 460 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10 ominf 9209 . . . 4 ¬ ω ∈ Fin
11 ensymfib 9138 . . . . . 6 (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
125, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13 breq2 5114 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
1413rspcev 3584 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15 isfi 8923 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
1614, 15sylibr 233 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1716ex 414 . . . . 5 (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
1812, 17sylbid 239 . . . 4 (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
1910, 18mtoi 198 . . 3 (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
2019adantl 483 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21 brsdom 8922 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
229, 20, 21sylanbrc 584 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107  wrex 3074  Vcvv 3448  wss 3915   class class class wbr 5110  Ord word 6321  ωcom 7807  cen 8887  cdom 8888  csdm 8889  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by:  isfiniteg  9255  infsdomnn  9256  infsdomnnOLD  9257  nnsdom  9597
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