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Theorem nnsdomg 9333
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 9685 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5369. (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 7886 . . . . . 6 Ord ω
2 ordelss 6390 . . . . . 6 ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
31, 2mpan 688 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
43adantr 479 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5 nnfi 9198 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ Fin)
6 ssdomfi2 9231 . . . . 5 ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
75, 6syl3an1 1160 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
84, 7mpd3an3 1458 . . 3 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
98ancoms 457 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10 ominf 9289 . . . 4 ¬ ω ∈ Fin
11 ensymfib 9218 . . . . . 6 (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
125, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13 breq2 5156 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
1413rspcev 3611 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15 isfi 9003 . . . . . . 7 (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
1614, 15sylibr 233 . . . . . 6 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
1716ex 411 . . . . 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 480 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21 brsdom 9002 . 2 (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
229, 20, 21sylanbrc 581 1 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098  wrex 3067  Vcvv 3473  wss 3949   class class class wbr 5152  Ord word 6373  ωcom 7876  cen 8967  cdom 8968  csdm 8969  Fincfn 8970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974
This theorem is referenced by:  isfiniteg  9335  infsdomnn  9336  infsdomnnOLD  9337  nnsdom  9685
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