Theorem nnsdomg 9298

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 9645 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5362. (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 7861	. . . . . 6 ⊢ Ord ω
2		ordelss 6377	. . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
3	1, 2	mpan 688	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
4	3	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω)
5		nnfi 9163	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
6		ssdomfi2 9196	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
7	5, 6	syl3an1 1163	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
8	4, 7	mpd3an3 1462	. . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
9	8	ancoms 459	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω)
10		ominf 9254	. . . 4 ⊢ ¬ ω ∈ Fin
11		ensymfib 9183	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
12	5, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴))
13		breq2 5151	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
14	13	rspcev 3612	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15		isfi 8968	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
16	14, 15	sylibr 233	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
17	16	ex 413	. . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
18	12, 17	sylbid 239	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
19	10, 18	mtoi 198	. . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
20	19	adantl 482	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21		brsdom 8967	. 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω))
22	9, 20, 21	sylanbrc 583	1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ⊆ wss 3947   class class class wbr 5147  Ord word 6360  ωcom 7851   ≈ cen 8932   ≼ cdom 8933   ≺ csdm 8934  Fincfn 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939

This theorem is referenced by:  isfiniteg  9300  infsdomnn  9301  infsdomnnOLD  9302  nnsdom  9645