naddwordnexlem4 - Mathbox for Richard Penner

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Theorem naddwordnexlem4 43721

Description: When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, there exists a product with omega such that the ordinal sum with 𝐴 is less than or equal to 𝐵 while the natural sum is larger than 𝐵. (Contributed by RP, 15-Feb-2025.)

**Hypotheses**
Ref	Expression
naddwordnex.a	⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀))
naddwordnex.b	⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁))
naddwordnex.c	⊢ (𝜑 → 𝐶 ∈ 𝐷)
naddwordnex.d	⊢ (𝜑 → 𝐷 ∈ On)
naddwordnex.m	⊢ (𝜑 → 𝑀 ∈ ω)
naddwordnex.n	⊢ (𝜑 → 𝑁 ∈ 𝑀)
naddwordnexlem4.s	⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}

**Assertion**
Ref	Expression
naddwordnexlem4	⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1_o)((𝐶 +_o 𝑥) = 𝐷 ∧ (𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o 𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶,𝑦 𝑥,𝐷,𝑦 𝑥,𝑆 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑦) 𝐵(𝑦) 𝑆(𝑦) 𝑀(𝑥,𝑦) 𝑁(𝑥,𝑦)

Proof of Theorem naddwordnexlem4 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		naddwordnexlem4.s	. . . . 5 ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}
2	1	ssrab3 4035	. . . 4 ⊢ 𝑆 ⊆ On
3		oveq2 7369	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝐶 +_o 𝑦) = (𝐶 +_o 𝐷))
4	3	sseq2d 3967	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝐷 ⊆ (𝐶 +_o 𝑦) ↔ 𝐷 ⊆ (𝐶 +_o 𝐷)))
5		naddwordnex.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ On)
6		naddwordnex.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		onelon 6343	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
8	5, 6, 7	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
9		oaword2 8483	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → 𝐷 ⊆ (𝐶 +_o 𝐷))
10	5, 8, 9	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ (𝐶 +_o 𝐷))
11	4, 5, 10	elrabd 3649	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
12	11, 1	eleqtrrdi 2848	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
13	12	ne0d 4295	. . . 4 ⊢ (𝜑 → 𝑆 ≠ ∅)
14		oninton 7743	. . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ On)
15	2, 13, 14	sylancr 588	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ On)
16		oveq2 7369	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (𝐶 +_o 𝑦) = (𝐶 +_o ∅))
17		oa0 8446	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (𝐶 +_o ∅) = 𝐶)
18	8, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 +_o ∅) = 𝐶)
19	16, 18	sylan9eqr 2794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝐶 +_o 𝑦) = 𝐶)
20	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 𝐶 ∈ 𝐷)
21	19, 20	eqeltrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝐶 +_o 𝑦) ∈ 𝐷)
22	21	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 = ∅ → (𝐶 +_o 𝑦) ∈ 𝐷))
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝑦 = ∅ → (𝐶 +_o 𝑦) ∈ 𝐷))
24	23	con3d 152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (¬ (𝐶 +_o 𝑦) ∈ 𝐷 → ¬ 𝑦 = ∅))
25		oacl 8465	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +_o 𝑦) ∈ On)
26	8, 25	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐶 +_o 𝑦) ∈ On)
27		ontri1 6352	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ (𝐶 +_o 𝑦) ∈ On) → (𝐷 ⊆ (𝐶 +_o 𝑦) ↔ ¬ (𝐶 +_o 𝑦) ∈ 𝐷))
28	5, 26, 27	syl2an2r 686	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐷 ⊆ (𝐶 +_o 𝑦) ↔ ¬ (𝐶 +_o 𝑦) ∈ 𝐷))
29		on0eln0 6375	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
30		df-ne 2934	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ¬ 𝑦 = ∅)
31	29, 30	bitrdi 287	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ ¬ 𝑦 = ∅))
32	31	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (∅ ∈ 𝑦 ↔ ¬ 𝑦 = ∅))
33	24, 28, 32	3imtr4d 294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐷 ⊆ (𝐶 +_o 𝑦) → ∅ ∈ 𝑦))
34	33	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ On → (𝐷 ⊆ (𝐶 +_o 𝑦) → ∅ ∈ 𝑦)))
35	34	ralrimiv 3128	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ On (𝐷 ⊆ (𝐶 +_o 𝑦) → ∅ ∈ 𝑦))
36		0ex 5253	. . . . . 6 ⊢ ∅ ∈ V
37	36	elintrab 4916	. . . . 5 ⊢ (∅ ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} ↔ ∀𝑦 ∈ On (𝐷 ⊆ (𝐶 +_o 𝑦) → ∅ ∈ 𝑦))
38	35, 37	sylibr 234	. . . 4 ⊢ (𝜑 → ∅ ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
39	1	inteqi 4907	. . . 4 ⊢ ∩ 𝑆 = ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}
40	38, 39	eleqtrrdi 2848	. . 3 ⊢ (𝜑 → ∅ ∈ ∩ 𝑆)
41		ondif1 8431	. . 3 ⊢ (∩ 𝑆 ∈ (On ∖ 1_o) ↔ (∩ 𝑆 ∈ On ∧ ∅ ∈ ∩ 𝑆))
42	15, 40, 41	sylanbrc 584	. 2 ⊢ (𝜑 → ∩ 𝑆 ∈ (On ∖ 1_o))
43		onzsl 7791	. . . . . 6 ⊢ (∩ 𝑆 ∈ On ↔ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
44	15, 43	sylib 218	. . . . 5 ⊢ (𝜑 → (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
45		oveq2 7369	. . . . . . . . 9 ⊢ (∩ 𝑆 = ∅ → (𝐶 +_o ∩ 𝑆) = (𝐶 +_o ∅))
46	45, 18	sylan9eqr 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → (𝐶 +_o ∩ 𝑆) = 𝐶)
47		onelpss 6358	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∈ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷)))
48	8, 5, 47	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷)))
49	6, 48	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷))
50	49	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → 𝐶 ⊆ 𝐷)
52	46, 51	eqsstrd 3969	. . . . . . 7 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷)
53	52	ex 412	. . . . . 6 ⊢ (𝜑 → (∩ 𝑆 = ∅ → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷))
54		oveq2 7369	. . . . . . . . 9 ⊢ (∩ 𝑆 = suc 𝑧 → (𝐶 +_o ∩ 𝑆) = (𝐶 +_o suc 𝑧))
55		oasuc 8454	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On) → (𝐶 +_o suc 𝑧) = suc (𝐶 +_o 𝑧))
56	8, 55	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐶 +_o suc 𝑧) = suc (𝐶 +_o 𝑧))
57	54, 56	sylan9eqr 2794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ ∩ 𝑆 = suc 𝑧) → (𝐶 +_o ∩ 𝑆) = suc (𝐶 +_o 𝑧))
58		vex 3445	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
59	58	sucid 6402	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ suc 𝑧
60		eleq2 2826	. . . . . . . . . . . 12 ⊢ (∩ 𝑆 = suc 𝑧 → (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ suc 𝑧))
61	59, 60	mpbiri 258	. . . . . . . . . . 11 ⊢ (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆)
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆))
63	39	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
64		oveq2 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝐶 +_o 𝑦) = (𝐶 +_o 𝑧))
65	64	sseq2d 3967	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐷 ⊆ (𝐶 +_o 𝑦) ↔ 𝐷 ⊆ (𝐶 +_o 𝑧)))
66	65	onnminsb 7747	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} → ¬ 𝐷 ⊆ (𝐶 +_o 𝑧)))
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} → ¬ 𝐷 ⊆ (𝐶 +_o 𝑧)))
68	63, 67	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ 𝑆 → ¬ 𝐷 ⊆ (𝐶 +_o 𝑧)))
69		oacl 8465	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On) → (𝐶 +_o 𝑧) ∈ On)
70	8, 69	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐶 +_o 𝑧) ∈ On)
71		ontri1 6352	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ (𝐶 +_o 𝑧) ∈ On) → (𝐷 ⊆ (𝐶 +_o 𝑧) ↔ ¬ (𝐶 +_o 𝑧) ∈ 𝐷))
72	5, 70, 71	syl2an2r 686	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐷 ⊆ (𝐶 +_o 𝑧) ↔ ¬ (𝐶 +_o 𝑧) ∈ 𝐷))
73	72	con2bid 354	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ((𝐶 +_o 𝑧) ∈ 𝐷 ↔ ¬ 𝐷 ⊆ (𝐶 +_o 𝑧)))
74	68, 73	sylibrd 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ 𝑆 → (𝐶 +_o 𝑧) ∈ 𝐷))
75		onsucss 43586	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ On → ((𝐶 +_o 𝑧) ∈ 𝐷 → suc (𝐶 +_o 𝑧) ⊆ 𝐷))
76	5, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐶 +_o 𝑧) ∈ 𝐷 → suc (𝐶 +_o 𝑧) ⊆ 𝐷))
77	76	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ((𝐶 +_o 𝑧) ∈ 𝐷 → suc (𝐶 +_o 𝑧) ⊆ 𝐷))
78	62, 74, 77	3syld 60	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∩ 𝑆 = suc 𝑧 → suc (𝐶 +_o 𝑧) ⊆ 𝐷))
79	78	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ ∩ 𝑆 = suc 𝑧) → suc (𝐶 +_o 𝑧) ⊆ 𝐷)
80	57, 79	eqsstrd 3969	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ ∩ 𝑆 = suc 𝑧) → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷)
81	80	rexlimdva2 3140	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷))
82		oalim 8462	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +_o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧))
83	8, 82	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +_o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧))
84		onelon 6343	. . . . . . . . . . . . . . 15 ⊢ ((∩ 𝑆 ∈ On ∧ 𝑧 ∈ ∩ 𝑆) → 𝑧 ∈ On)
85	15, 84	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ∩ 𝑆) → 𝑧 ∈ On)
86	85	ex 412	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ ∩ 𝑆 → 𝑧 ∈ On))
87	86	ancrd 551	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ ∩ 𝑆 → (𝑧 ∈ On ∧ 𝑧 ∈ ∩ 𝑆)))
88	74	expimpd 453	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑧 ∈ On ∧ 𝑧 ∈ ∩ 𝑆) → (𝐶 +_o 𝑧) ∈ 𝐷))
89		onelss 6360	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ On → ((𝐶 +_o 𝑧) ∈ 𝐷 → (𝐶 +_o 𝑧) ⊆ 𝐷))
90	5, 89	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 +_o 𝑧) ∈ 𝐷 → (𝐶 +_o 𝑧) ⊆ 𝐷))
91	87, 88, 90	3syld 60	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ ∩ 𝑆 → (𝐶 +_o 𝑧) ⊆ 𝐷))
92	91	ralrimiv 3128	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧) ⊆ 𝐷)
93		iunss 5001	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧) ⊆ 𝐷 ↔ ∀𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧) ⊆ 𝐷)
94	92, 93	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧) ⊆ 𝐷)
95	94	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +_o 𝑧) ⊆ 𝐷)
96	83, 95	eqsstrd 3969	. . . . . . 7 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷)
97	96	ex 412	. . . . . 6 ⊢ (𝜑 → ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷))
98	53, 81, 97	3jaod 1432	. . . . 5 ⊢ (𝜑 → ((∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷))
99	44, 98	mpd 15	. . . 4 ⊢ (𝜑 → (𝐶 +_o ∩ 𝑆) ⊆ 𝐷)
100	4	rspcev 3577	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐷 ⊆ (𝐶 +_o 𝐷)) → ∃𝑦 ∈ On 𝐷 ⊆ (𝐶 +_o 𝑦))
101	5, 10, 100	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ On 𝐷 ⊆ (𝐶 +_o 𝑦))
102		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑦𝐷
103		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
104		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑦 +_o
105		nfrab1 3420	. . . . . . . . . 10 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}
106	105	nfint 4913	. . . . . . . . 9 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}
107	103, 104, 106	nfov 7391	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
108	102, 107	nfss 3927	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
109		oveq2 7369	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} → (𝐶 +_o 𝑦) = (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}))
110	109	sseq2d 3967	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} → (𝐷 ⊆ (𝐶 +_o 𝑦) ↔ 𝐷 ⊆ (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})))
111	108, 110	onminsb 7742	. . . . . 6 ⊢ (∃𝑦 ∈ On 𝐷 ⊆ (𝐶 +_o 𝑦) → 𝐷 ⊆ (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}))
112	101, 111	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)}))
113	39	oveq2i 7372	. . . . 5 ⊢ (𝐶 +_o ∩ 𝑆) = (𝐶 +_o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)})
114	112, 113	sseqtrrdi 3976	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ (𝐶 +_o ∩ 𝑆))
115	99, 114	eqssd 3952	. . 3 ⊢ (𝜑 → (𝐶 +_o ∩ 𝑆) = 𝐷)
116		omelon 9560	. . . . . 6 ⊢ ω ∈ On
117		omcl 8466	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·_o 𝐷) ∈ On)
118	116, 5, 117	sylancr 588	. . . . 5 ⊢ (𝜑 → (ω ·_o 𝐷) ∈ On)
119	116	a1i 11	. . . . . . 7 ⊢ (𝜑 → ω ∈ On)
120		naddwordnex.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑀)
121		naddwordnex.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ω)
122	120, 121	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
123		ontr1 6365	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
124	119, 122, 123	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
125		nnon 7817	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
126	124, 125	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
127		oaword1 8482	. . . . 5 ⊢ (((ω ·_o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·_o 𝐷) ⊆ ((ω ·_o 𝐷) +_o 𝑁))
128	118, 126, 127	syl2anc 585	. . . 4 ⊢ (𝜑 → (ω ·_o 𝐷) ⊆ ((ω ·_o 𝐷) +_o 𝑁))
129		naddwordnex.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀))
130	129	oveq1d 7376	. . . . 5 ⊢ (𝜑 → (𝐴 +_o (ω ·_o ∩ 𝑆)) = (((ω ·_o 𝐶) +_o 𝑀) +_o (ω ·_o ∩ 𝑆)))
131		omcl 8466	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·_o 𝐶) ∈ On)
132	116, 8, 131	sylancr 588	. . . . . 6 ⊢ (𝜑 → (ω ·_o 𝐶) ∈ On)
133		nnon 7817	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On)
134	121, 133	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ On)
135		omcl 8466	. . . . . . 7 ⊢ ((ω ∈ On ∧ ∩ 𝑆 ∈ On) → (ω ·_o ∩ 𝑆) ∈ On)
136	116, 15, 135	sylancr 588	. . . . . 6 ⊢ (𝜑 → (ω ·_o ∩ 𝑆) ∈ On)
137		oaass 8491	. . . . . 6 ⊢ (((ω ·_o 𝐶) ∈ On ∧ 𝑀 ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On) → (((ω ·_o 𝐶) +_o 𝑀) +_o (ω ·_o ∩ 𝑆)) = ((ω ·_o 𝐶) +_o (𝑀 +_o (ω ·_o ∩ 𝑆))))
138	132, 134, 136, 137	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (((ω ·_o 𝐶) +_o 𝑀) +_o (ω ·_o ∩ 𝑆)) = ((ω ·_o 𝐶) +_o (𝑀 +_o (ω ·_o ∩ 𝑆))))
139	15, 116	jctil 519	. . . . . . . . 9 ⊢ (𝜑 → (ω ∈ On ∧ ∩ 𝑆 ∈ On))
140		omword1 8503	. . . . . . . . 9 ⊢ (((ω ∈ On ∧ ∩ 𝑆 ∈ On) ∧ ∅ ∈ ∩ 𝑆) → ω ⊆ (ω ·_o ∩ 𝑆))
141	139, 40, 140	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ω ⊆ (ω ·_o ∩ 𝑆))
142		oaabs 8579	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ (ω ·_o ∩ 𝑆) ∈ On) ∧ ω ⊆ (ω ·_o ∩ 𝑆)) → (𝑀 +_o (ω ·_o ∩ 𝑆)) = (ω ·_o ∩ 𝑆))
143	121, 136, 141, 142	syl21anc 838	. . . . . . 7 ⊢ (𝜑 → (𝑀 +_o (ω ·_o ∩ 𝑆)) = (ω ·_o ∩ 𝑆))
144	143	oveq2d 7377	. . . . . 6 ⊢ (𝜑 → ((ω ·_o 𝐶) +_o (𝑀 +_o (ω ·_o ∩ 𝑆))) = ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)))
145		odi 8509	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On ∧ ∩ 𝑆 ∈ On) → (ω ·_o (𝐶 +_o ∩ 𝑆)) = ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)))
146	119, 8, 15, 145	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (ω ·_o (𝐶 +_o ∩ 𝑆)) = ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)))
147	115	oveq2d 7377	. . . . . 6 ⊢ (𝜑 → (ω ·_o (𝐶 +_o ∩ 𝑆)) = (ω ·_o 𝐷))
148	144, 146, 147	3eqtr2d 2778	. . . . 5 ⊢ (𝜑 → ((ω ·_o 𝐶) +_o (𝑀 +_o (ω ·_o ∩ 𝑆))) = (ω ·_o 𝐷))
149	130, 138, 148	3eqtrd 2776	. . . 4 ⊢ (𝜑 → (𝐴 +_o (ω ·_o ∩ 𝑆)) = (ω ·_o 𝐷))
150		naddwordnex.b	. . . 4 ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁))
151	128, 149, 150	3sstr4d 3990	. . 3 ⊢ (𝜑 → (𝐴 +_o (ω ·_o ∩ 𝑆)) ⊆ 𝐵)
152		naddcl 8608	. . . . . . . 8 ⊢ (((ω ·_o 𝐶) ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On) → ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) ∈ On)
153	132, 136, 152	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) ∈ On)
154	118, 153, 134	3jca 1129	. . . . . 6 ⊢ (𝜑 → ((ω ·_o 𝐷) ∈ On ∧ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) ∈ On ∧ 𝑀 ∈ On))
155	147, 146	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → (ω ·_o 𝐷) = ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)))
156		naddgeoa 43714	. . . . . . . 8 ⊢ (((ω ·_o 𝐶) ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On) → ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)) ⊆ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)))
157	132, 136, 156	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((ω ·_o 𝐶) +_o (ω ·_o ∩ 𝑆)) ⊆ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)))
158	155, 157	eqsstrd 3969	. . . . . 6 ⊢ (𝜑 → (ω ·_o 𝐷) ⊆ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)))
159		oawordri 8480	. . . . . 6 ⊢ (((ω ·_o 𝐷) ∈ On ∧ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) ∈ On ∧ 𝑀 ∈ On) → ((ω ·_o 𝐷) ⊆ ((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) → ((ω ·_o 𝐷) +_o 𝑀) ⊆ (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀)))
160	154, 158, 159	sylc 65	. . . . 5 ⊢ (𝜑 → ((ω ·_o 𝐷) +_o 𝑀) ⊆ (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀))
161		naddonnn 43715	. . . . . . . . 9 ⊢ (((ω ·_o 𝐶) ∈ On ∧ 𝑀 ∈ ω) → ((ω ·_o 𝐶) +_o 𝑀) = ((ω ·_o 𝐶) +no 𝑀))
162	132, 121, 161	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((ω ·_o 𝐶) +_o 𝑀) = ((ω ·_o 𝐶) +no 𝑀))
163	129, 162	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +no 𝑀))
164	163	oveq1d 7376	. . . . . 6 ⊢ (𝜑 → (𝐴 +no (ω ·_o ∩ 𝑆)) = (((ω ·_o 𝐶) +no 𝑀) +no (ω ·_o ∩ 𝑆)))
165		naddass 8627	. . . . . . . 8 ⊢ (((ω ·_o 𝐶) ∈ On ∧ 𝑀 ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On) → (((ω ·_o 𝐶) +no 𝑀) +no (ω ·_o ∩ 𝑆)) = ((ω ·_o 𝐶) +no (𝑀 +no (ω ·_o ∩ 𝑆))))
166	132, 134, 136, 165	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (((ω ·_o 𝐶) +no 𝑀) +no (ω ·_o ∩ 𝑆)) = ((ω ·_o 𝐶) +no (𝑀 +no (ω ·_o ∩ 𝑆))))
167		naddcom 8613	. . . . . . . . 9 ⊢ ((𝑀 ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On) → (𝑀 +no (ω ·_o ∩ 𝑆)) = ((ω ·_o ∩ 𝑆) +no 𝑀))
168	134, 136, 167	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝑀 +no (ω ·_o ∩ 𝑆)) = ((ω ·_o ∩ 𝑆) +no 𝑀))
169	168	oveq2d 7377	. . . . . . 7 ⊢ (𝜑 → ((ω ·_o 𝐶) +no (𝑀 +no (ω ·_o ∩ 𝑆))) = ((ω ·_o 𝐶) +no ((ω ·_o ∩ 𝑆) +no 𝑀)))
170		naddonnn 43715	. . . . . . . . 9 ⊢ ((((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) ∈ On ∧ 𝑀 ∈ ω) → (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀) = (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +no 𝑀))
171	153, 121, 170	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀) = (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +no 𝑀))
172		naddass 8627	. . . . . . . . 9 ⊢ (((ω ·_o 𝐶) ∈ On ∧ (ω ·_o ∩ 𝑆) ∈ On ∧ 𝑀 ∈ On) → (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +no 𝑀) = ((ω ·_o 𝐶) +no ((ω ·_o ∩ 𝑆) +no 𝑀)))
173	132, 136, 134, 172	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +no 𝑀) = ((ω ·_o 𝐶) +no ((ω ·_o ∩ 𝑆) +no 𝑀)))
174	171, 173	eqtr2d 2773	. . . . . . 7 ⊢ (𝜑 → ((ω ·_o 𝐶) +no ((ω ·_o ∩ 𝑆) +no 𝑀)) = (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀))
175	166, 169, 174	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (((ω ·_o 𝐶) +no 𝑀) +no (ω ·_o ∩ 𝑆)) = (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀))
176	164, 175	eqtr2d 2773	. . . . 5 ⊢ (𝜑 → (((ω ·_o 𝐶) +no (ω ·_o ∩ 𝑆)) +_o 𝑀) = (𝐴 +no (ω ·_o ∩ 𝑆)))
177	160, 176	sseqtrd 3971	. . . 4 ⊢ (𝜑 → ((ω ·_o 𝐷) +_o 𝑀) ⊆ (𝐴 +no (ω ·_o ∩ 𝑆)))
178	134, 118	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ On ∧ (ω ·_o 𝐷) ∈ On))
179		oaordi 8476	. . . . . 6 ⊢ ((𝑀 ∈ On ∧ (ω ·_o 𝐷) ∈ On) → (𝑁 ∈ 𝑀 → ((ω ·_o 𝐷) +_o 𝑁) ∈ ((ω ·_o 𝐷) +_o 𝑀)))
180	178, 120, 179	sylc 65	. . . . 5 ⊢ (𝜑 → ((ω ·_o 𝐷) +_o 𝑁) ∈ ((ω ·_o 𝐷) +_o 𝑀))
181	150, 180	eqeltrd 2837	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ((ω ·_o 𝐷) +_o 𝑀))
182	177, 181	sseldd 3935	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 +no (ω ·_o ∩ 𝑆)))
183	115, 151, 182	3jca 1129	. 2 ⊢ (𝜑 → ((𝐶 +_o ∩ 𝑆) = 𝐷 ∧ (𝐴 +_o (ω ·_o ∩ 𝑆)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o ∩ 𝑆))))
184		oveq2 7369	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐶 +_o 𝑥) = (𝐶 +_o ∩ 𝑆))
185	184	eqeq1d 2739	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → ((𝐶 +_o 𝑥) = 𝐷 ↔ (𝐶 +_o ∩ 𝑆) = 𝐷))
186		oveq2 7369	. . . . . 6 ⊢ (𝑥 = ∩ 𝑆 → (ω ·_o 𝑥) = (ω ·_o ∩ 𝑆))
187	186	oveq2d 7377	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +_o (ω ·_o 𝑥)) = (𝐴 +_o (ω ·_o ∩ 𝑆)))
188	187	sseq1d 3966	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → ((𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ↔ (𝐴 +_o (ω ·_o ∩ 𝑆)) ⊆ 𝐵))
189	186	oveq2d 7377	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +no (ω ·_o 𝑥)) = (𝐴 +no (ω ·_o ∩ 𝑆)))
190	189	eleq2d 2823	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → (𝐵 ∈ (𝐴 +no (ω ·_o 𝑥)) ↔ 𝐵 ∈ (𝐴 +no (ω ·_o ∩ 𝑆))))
191	185, 188, 190	3anbi123d 1439	. . 3 ⊢ (𝑥 = ∩ 𝑆 → (((𝐶 +_o 𝑥) = 𝐷 ∧ (𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o 𝑥))) ↔ ((𝐶 +_o ∩ 𝑆) = 𝐷 ∧ (𝐴 +_o (ω ·_o ∩ 𝑆)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o ∩ 𝑆)))))
192	191	rspcev 3577	. 2 ⊢ ((∩ 𝑆 ∈ (On ∖ 1_o) ∧ ((𝐶 +_o ∩ 𝑆) = 𝐷 ∧ (𝐴 +_o (ω ·_o ∩ 𝑆)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o ∩ 𝑆)))) → ∃𝑥 ∈ (On ∖ 1_o)((𝐶 +_o 𝑥) = 𝐷 ∧ (𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o 𝑥))))
193	42, 183, 192	syl2anc 585	1 ⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1_o)((𝐶 +_o 𝑥) = 𝐷 ∧ (𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 {crab 3400 Vcvv 3441 ∖ cdif 3899 ⊆ wss 3902 ∅c0 4286 ∩ cint 4903 ∪ ciun 4947 Oncon0 6318 Lim wlim 6319 suc csuc 6320 (class class class)co 7361 ωcom 7811 1_oc1o 8393 +_o coa 8397 ·_o comu 8398 +no cnadd 8596

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-inf2 9555

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-oadd 8404 df-omul 8405 df-nadd 8597

This theorem is referenced by: (None)

W3C validator