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Theorem nnsdomg 9332
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 9691 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5370. (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 7896 . . . . . 6 Ord ω
2 ordelss 6401 . . . . . 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 9205 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ Fin)
6 ssdomfi2 9234 . . . . 5 ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
75, 6syl3an1 1162 . . . 4 ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
84, 7mpd3an3 1461 . . 3 ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
98ancoms 458 . 2 ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10 ominf 9291 . . . 4 ¬ ω ∈ Fin
11 ensymfib 9221 . . . . . 6 (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
125, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13 breq2 5151 . . . . . . . 8 (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
1413rspcev 3621 . . . . . . 7 ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15 isfi 9014 . . . . . . 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 9013 . 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 2105  wrex 3067  Vcvv 3477  wss 3962   class class class wbr 5147  Ord word 6384  ωcom 7886  cen 8980  cdom 8981  csdm 8982  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987
This theorem is referenced by:  isfiniteg  9334  infsdomnn  9335  infsdomnnOLD  9336  nnsdom  9691
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