Theorem nnoeomeqom 43305

Description: Any natural number at least as large as two raised to the power of omega is omega. Lemma 3.25 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
nnoeomeqom	⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)

Proof of Theorem nnoeomeqom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ ω)
2		nnon 7805	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ On)
4		omelon 9542	. . . . 5 ⊢ ω ∈ On
5		limom 7815	. . . . 5 ⊢ Lim ω
6	4, 5	pm3.2i 470	. . . 4 ⊢ (ω ∈ On ∧ Lim ω)
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (ω ∈ On ∧ Lim ω))
8		0elon 6362	. . . . 5 ⊢ ∅ ∈ On
9	8	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ∈ On)
10		0ss 4351	. . . . 5 ⊢ ∅ ⊆ 1o
11	10	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ⊆ 1o)
12		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 1o ∈ 𝐴)
13		ontr2 6355	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → ((∅ ⊆ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
14	13	imp 406	. . . 4 ⊢ (((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆ 1o ∧ 1o ∈ 𝐴)) → ∅ ∈ 𝐴)
15	9, 3, 11, 12, 14	syl22anc 838	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)
16		oelim 8452	. . 3 ⊢ (((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥))
17	3, 7, 15, 16	syl21anc 837	. 2 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥))
18		ovex 7382	. . . 4 ⊢ (𝐴 ↑o 𝑥) ∈ V
19	18	dfiun2 4982	. . 3 ⊢ ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)}
20		eluniab 4872	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)))
21		19.42v 1953	. . . . . . . 8 ⊢ (∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
22		3anass 1094	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
23	22	exbii 1848	. . . . . . . 8 ⊢ (∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
24		df-rex 3054	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥) ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
25	24	anbi2i 623	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
26	21, 23, 25	3bitr4ri 304	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
27	26	exbii 1848	. . . . . 6 ⊢ (∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑦∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
28		excom 2163	. . . . . 6 ⊢ (∃𝑦∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
29	20, 27, 28	3bitri 297	. . . . 5 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
30		simpr3 1197	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 = (𝐴 ↑o 𝑥))
31		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑥 ∈ ω)
32		nnecl 8531	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ↑o 𝑥) ∈ ω)
33	1, 31, 32	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ∈ ω)
34		onelss 6349	. . . . . . . . . . 11 ⊢ (ω ∈ On → ((𝐴 ↑o 𝑥) ∈ ω → (𝐴 ↑o 𝑥) ⊆ ω))
35	4, 33, 34	mpsyl 68	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ⊆ ω)
36	30, 35	eqsstrd 3970	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 ⊆ ω)
37		simpr1 1195	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ 𝑦)
38	36, 37	sseldd 3936	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ ω)
39	38	ex 412	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω))
40	39	exlimdvv 1934	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω))
41		peano2 7823	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
42	41	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ∈ ω)
43		ovex 7382	. . . . . . . . . 10 ⊢ (𝐴 ↑o suc 𝑧) ∈ V
44	43	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → (𝐴 ↑o suc 𝑧) ∈ V)
45	2	anim1i 615	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ∈ On ∧ 1o ∈ 𝐴))
46		ondif2 8420	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
47	45, 46	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ (On ∖ 2o))
48		nnon 7805	. . . . . . . . . . . . 13 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
49	41, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ On)
50		oeworde 8511	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝑧 ∈ On) → suc 𝑧 ⊆ (𝐴 ↑o suc 𝑧))
51	47, 49, 50	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ⊆ (𝐴 ↑o suc 𝑧))
52		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
53	52	sucid 6391	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ suc 𝑧
54	53	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
55	51, 54	sseldd 3936	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴 ↑o suc 𝑧))
56		eqidd 2730	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧))
57	55, 42, 56	3jca 1128	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → (𝑧 ∈ (𝐴 ↑o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧)))
58		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ (𝐴 ↑o suc 𝑧)))
59		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑦 = (𝐴 ↑o suc 𝑧) ↔ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧)))
60	58, 59	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → ((𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)) ↔ (𝑧 ∈ (𝐴 ↑o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧))))
61	44, 57, 60	spcedv 3553	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)))
62		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑧))
64	63	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → (𝑦 = (𝐴 ↑o 𝑥) ↔ 𝑦 = (𝐴 ↑o suc 𝑧)))
65	62, 64	3anbi23d 1441	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑧 → ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧))))
66	65	exbidv 1921	. . . . . . . 8 ⊢ (𝑥 = suc 𝑧 → (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧))))
67	42, 61, 66	spcedv 3553	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
68	67	ex 412	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝑧 ∈ ω → ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
69	40, 68	impbid 212	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ 𝑧 ∈ ω))
70	29, 69	bitrid 283	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ 𝑧 ∈ ω))
71	70	eqrdv 2727	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} = ω)
72	19, 71	eqtrid 2776	. 2 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ω)
73	17, 72	eqtrd 2764	1 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  ∪ cuni 4858  ∪ ciun 4941  Oncon0 6307  Lim wlim 6308  suc csuc 6309  (class class class)co 7349  ωcom 7799  1_oc1o 8381  2_oc2o 8382   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-oexp 8394

This theorem is referenced by:  oenord1ex  43308  oaomoencom  43310