Theorem nnoeomeqom 43966

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 487	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ ω)
2		nnon 7868	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 18	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ On)
4		omelon 9615	. . . . 5 ⊢ ω ∈ On
5		limom 7878	. . . . 5 ⊢ Lim ω
6	4, 5	pm3.2i 475	. . . 4 ⊢ (ω ∈ On ∧ Lim ω)
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (ω ∈ On ∧ Lim ω))
8		0elon 6417	. . . . 5 ⊢ ∅ ∈ On
9	8	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ∈ On)
10		0ss 4362	. . . . 5 ⊢ ∅ ⊆ 1o
11	10	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ⊆ 1o)
12		simpr 489	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 1o ∈ 𝐴)
13		ontr2 6410	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → ((∅ ⊆ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴))
14	13	imp 411	. . . 4 ⊢ (((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆ 1o ∧ 1o ∈ 𝐴)) → ∅ ∈ 𝐴)
15	9, 3, 11, 12, 14	syl22anc 851	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)
16		oelim 8519	. . 3 ⊢ (((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥))
17	3, 7, 15, 16	syl21anc 850	. 2 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥))
18		ovex 7444	. . . 4 ⊢ (𝐴 ↑o 𝑥) ∈ V
19	18	dfiun2 4998	. . 3 ⊢ ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)}
20		eluniab 4888	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)))
21		19.42v 1980	. . . . . . . 8 ⊢ (∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
22		3anass 1109	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
23	22	exbii 1875	. . . . . . . 8 ⊢ (∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
24		df-rex 3096	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥) ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
25	24	anbi2i 634	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
26	21, 23, 25	3bitr4ri 307	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
27	26	exbii 1875	. . . . . 6 ⊢ (∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑦∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
28		excom 2203	. . . . . 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 300	. . . . 5 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
30		simpr3 1213	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 = (𝐴 ↑o 𝑥))
31		simp2 1153	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑥 ∈ ω)
32		nnecl 8599	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ↑o 𝑥) ∈ ω)
33	1, 31, 32	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ∈ ω)
34		onelss 6404	. . . . . . . . . . 11 ⊢ (ω ∈ On → ((𝐴 ↑o 𝑥) ∈ ω → (𝐴 ↑o 𝑥) ⊆ ω))
35	4, 33, 34	mpsyl 69	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ⊆ ω)
36	30, 35	eqsstrd 3977	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 ⊆ ω)
37		simpr1 1211	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ 𝑦)
38	36, 37	sseldd 3944	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ ω)
39	38	ex 417	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω))
40	39	exlimdvv 1961	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω))
41		peano2 7886	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
42	41	adantl 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ∈ ω)
43		ovex 7444	. . . . . . . . . 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 626	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ∈ On ∧ 1o ∈ 𝐴))
46		ondif2 8487	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
47	45, 46	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → 𝐴 ∈ (On ∖ 2o))
48		nnon 7868	. . . . . . . . . . . . 13 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
49	41, 48	syl 18	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ On)
50		oeworde 8579	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝑧 ∈ On) → suc 𝑧 ⊆ (𝐴 ↑o suc 𝑧))
51	47, 49, 50	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ⊆ (𝐴 ↑o suc 𝑧))
52		vex 3465	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
53	52	sucid 6446	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ suc 𝑧
54	53	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
55	51, 54	sseldd 3944	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴 ↑o suc 𝑧))
56		eqidd 2770	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧))
57	55, 42, 56	3jca 1144	. . . . . . . . 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 2858	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ (𝐴 ↑o suc 𝑧)))
59		eqeq1 2773	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑦 = (𝐴 ↑o suc 𝑧) ↔ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧)))
60	58, 59	3anbi13d 1464	. . . . . . . . 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 3564	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)))
62		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑧))
64	63	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → (𝑦 = (𝐴 ↑o 𝑥) ↔ 𝑦 = (𝐴 ↑o suc 𝑧)))
65	62, 64	3anbi23d 1465	. . . . . . . . 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 1948	. . . . . . . 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 3564	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 1o ∈ 𝐴) ∧ 𝑧 ∈ ω) → ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))
68	67	ex 417	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝑧 ∈ ω → ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))))
69	40, 68	impbid 215	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ 𝑧 ∈ ω))
70	29, 69	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ 𝑧 ∈ ω))
71	70	eqrdv 2767	. . 3 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} = ω)
72	19, 71	eqtrid 2816	. 2 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ω)
73	17, 72	eqtrd 2804	1 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∃wrex 3095  Vcvv 3461   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4874  ∪ ciun 4958  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7411  ωcom 7862  1_oc1o 8446  2_oc2o 8447   ↑_o coe 8452

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-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-inf2 9610

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 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  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-pred 6303  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-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-omul 8458  df-oexp 8459

This theorem is referenced by:  oenord1ex  43969  oaomoencom  43971