Theorem nnsdomg 9199

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 9566 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5294. (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 7816	. . . . . 6 ⊢ Ord ω
2		ordelss 6326	. . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
3	1, 2	mpan 696	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
4	3	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5		nnfi 9092	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
6		ssdomfi2 9121	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
7	5, 6	syl3an1 1169	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
8	4, 7	mpd3an3 1470	. . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
9	8	ancoms 459	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10		ominf 9164	. . . 4 ⊢ ¬ ω ∈ Fin
11		ensymfib 9108	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
12	5, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13		breq2 5076	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
14	13	rspcev 3560	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15		isfi 8912	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
16	14, 15	sylibr 235	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
17	16	ex 413	. . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
18	12, 17	sylbid 241	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
19	10, 18	mtoi 200	. . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
20	19	adantl 482	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21		brsdom 8911	. 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
22	9, 20, 21	sylanbrc 589	1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883   class class class wbr 5072  Ord word 6309  ωcom 7806   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by:  isfiniteg  9200  infsdomnn  9201  nnsdom  9566