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 9563 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5310. (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 7818	. . . . . 6 ⊢ Ord ω
2		ordelss 6333	. . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω)
3	1, 2	mpan 690	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
4	3	adantr 480	. . . 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 1163	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
8	4, 7	mpd3an3 1464	. . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω)
9	8	ancoms 458	. 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 5102	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴))
14	13	rspcev 3576	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥)
15		isfi 8912	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥)
16	14, 15	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin)
17	16	ex 412	. . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin))
18	12, 17	sylbid 240	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin))
19	10, 18	mtoi 199	. . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω)
20	19	adantl 481	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω)
21		brsdom 8911	. 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 206   ∧ wa 395   ∈ wcel 2113  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098  Ord word 6316  ωcom 7808   ≈ 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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by:  isfiniteg  9200  infsdomnn  9201  nnsdom  9563