Theorem nnsdomg 9258

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 9622 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5337. (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.

Step	Hyp	Ref	Expression
1		ordom 7871	. . . . . 6 ⊢ Ord ω
2		ordelss 6377	. . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
3	1, 2	mpan 702	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
4	3	adantr 485	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5		nnfi 9151	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
6		ssdomfi2 9180	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
7	5, 6	syl3an1 1179	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
8	4, 7	mpd3an3 1488	. . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
9	8	ancoms 463	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10		ominf 9223	. . . 4 ⊢ ¬ ω ∈ Fin
11		ensymfib 9167	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
12	5, 11	syl 18	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13		breq2 5117	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
14	13	rspcev 3590	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15		isfi 8971	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
16	14, 15	sylibr 237	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
17	16	ex 417	. . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
18	12, 17	sylbid 243	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
19	10, 18	mtoi 202	. . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
20	19	adantl 486	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21		brsdom 8970	. 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
22	9, 20, 21	sylanbrc 594	1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913   class class class wbr 5113  Ord word 6360  ωcom 7861   ≈ cen 8939   ≼ cdom 8940   ≺ csdm 8941  Fincfn 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7862  df-1o 8452  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946

This theorem is referenced by:  isfiniteg  9259  infsdomnn  9260  nnsdom  9622