Theorem naddwordnexlem4 42137

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 ∖ 1o)((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 4079	. . . 4 ⊢ 𝑆 ⊆ On
3		oveq2 7413	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝐶 +o 𝑦) = (𝐶 +o 𝐷))
4	3	sseq2d 4013	. . . . . . 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 6386	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
8	5, 6, 7	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
9		oaword2 8549	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → 𝐷 ⊆ (𝐶 +o 𝐷))
10	5, 8, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ (𝐶 +o 𝐷))
11	4, 5, 10	elrabd 3684	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
12	11, 1	eleqtrrdi 2844	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
13	12	ne0d 4334	. . . 4 ⊢ (𝜑 → 𝑆 ≠ ∅)
14		oninton 7779	. . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ On)
15	2, 13, 14	sylancr 587	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ On)
16		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (𝐶 +o 𝑦) = (𝐶 +o ∅))
17		oa0 8512	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = ∅) → 𝐶 ∈ 𝐷)
21	19, 20	eqeltrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝐶 +o 𝑦) ∈ 𝐷)
22	21	ex 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 = ∅ → (𝐶 +o 𝑦) ∈ 𝐷))
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝑦 = ∅ → (𝐶 +o 𝑦) ∈ 𝐷))
24	23	con3d 152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (¬ (𝐶 +o 𝑦) ∈ 𝐷 → ¬ 𝑦 = ∅))
25		oacl 8531	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o 𝑦) ∈ On)
26	8, 25	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐶 +o 𝑦) ∈ On)
27		ontri1 6395	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ (𝐶 +o 𝑦) ∈ On) → (𝐷 ⊆ (𝐶 +o 𝑦) ↔ ¬ (𝐶 +o 𝑦) ∈ 𝐷))
28	5, 26, 27	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐷 ⊆ (𝐶 +o 𝑦) ↔ ¬ (𝐶 +o 𝑦) ∈ 𝐷))
29		on0eln0 6417	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
30		df-ne 2941	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ¬ 𝑦 = ∅)
31	29, 30	bitrdi 286	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ ¬ 𝑦 = ∅))
32	31	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (∅ ∈ 𝑦 ↔ ¬ 𝑦 = ∅))
33	24, 28, 32	3imtr4d 293	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ On) → (𝐷 ⊆ (𝐶 +o 𝑦) → ∅ ∈ 𝑦))
34	33	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ On → (𝐷 ⊆ (𝐶 +o 𝑦) → ∅ ∈ 𝑦)))
35	34	ralrimiv 3145	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ On (𝐷 ⊆ (𝐶 +o 𝑦) → ∅ ∈ 𝑦))
36		0ex 5306	. . . . . 6 ⊢ ∅ ∈ V
37	36	elintrab 4963	. . . . 5 ⊢ (∅ ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)} ↔ ∀𝑦 ∈ On (𝐷 ⊆ (𝐶 +o 𝑦) → ∅ ∈ 𝑦))
38	35, 37	sylibr 233	. . . 4 ⊢ (𝜑 → ∅ ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
39	1	inteqi 4953	. . . 4 ⊢ ∩ 𝑆 = ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}
40	38, 39	eleqtrrdi 2844	. . 3 ⊢ (𝜑 → ∅ ∈ ∩ 𝑆)
41		ondif1 8497	. . 3 ⊢ (∩ 𝑆 ∈ (On ∖ 1o) ↔ (∩ 𝑆 ∈ On ∧ ∅ ∈ ∩ 𝑆))
42	15, 40, 41	sylanbrc 583	. 2 ⊢ (𝜑 → ∩ 𝑆 ∈ (On ∖ 1o))
43		onzsl 7831	. . . . . 6 ⊢ (∩ 𝑆 ∈ On ↔ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
44	15, 43	sylib 217	. . . . 5 ⊢ (𝜑 → (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
45		oveq2 7413	. . . . . . . . 9 ⊢ (∩ 𝑆 = ∅ → (𝐶 +o ∩ 𝑆) = (𝐶 +o ∅))
46	45, 18	sylan9eqr 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → (𝐶 +o ∩ 𝑆) = 𝐶)
47		onelpss 6401	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∈ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷)))
48	8, 5, 47	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷)))
49	6, 48	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ∧ 𝐶 ≠ 𝐷))
50	49	simpld 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
51	50	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → 𝐶 ⊆ 𝐷)
52	46, 51	eqsstrd 4019	. . . . . . 7 ⊢ ((𝜑 ∧ ∩ 𝑆 = ∅) → (𝐶 +o ∩ 𝑆) ⊆ 𝐷)
53	52	ex 413	. . . . . 6 ⊢ (𝜑 → (∩ 𝑆 = ∅ → (𝐶 +o ∩ 𝑆) ⊆ 𝐷))
54		oveq2 7413	. . . . . . . . 9 ⊢ (∩ 𝑆 = suc 𝑧 → (𝐶 +o ∩ 𝑆) = (𝐶 +o suc 𝑧))
55		oasuc 8520	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On) → (𝐶 +o suc 𝑧) = suc (𝐶 +o 𝑧))
56	8, 55	sylan 580	. . . . . . . . 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 3478	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
59	58	sucid 6443	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ suc 𝑧
60		eleq2 2822	. . . . . . . . . . . 12 ⊢ (∩ 𝑆 = suc 𝑧 → (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ suc 𝑧))
61	59, 60	mpbiri 257	. . . . . . . . . . 11 ⊢ (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆)
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆))
63	39	eleq2i 2825	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
64		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝐶 +o 𝑦) = (𝐶 +o 𝑧))
65	64	sseq2d 4013	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐷 ⊆ (𝐶 +o 𝑦) ↔ 𝐷 ⊆ (𝐶 +o 𝑧)))
66	65	onnminsb 7783	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)} → ¬ 𝐷 ⊆ (𝐶 +o 𝑧)))
67	66	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)} → ¬ 𝐷 ⊆ (𝐶 +o 𝑧)))
68	63, 67	biimtrid 241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ 𝑆 → ¬ 𝐷 ⊆ (𝐶 +o 𝑧)))
69		oacl 8531	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On) → (𝐶 +o 𝑧) ∈ On)
70	8, 69	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐶 +o 𝑧) ∈ On)
71		ontri1 6395	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ (𝐶 +o 𝑧) ∈ On) → (𝐷 ⊆ (𝐶 +o 𝑧) ↔ ¬ (𝐶 +o 𝑧) ∈ 𝐷))
72	5, 70, 71	syl2an2r 683	. . . . . . . . . . . 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 258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝑧 ∈ ∩ 𝑆 → (𝐶 +o 𝑧) ∈ 𝐷))
75		onsucss 42001	. . . . . . . . . . . 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 481	. . . . . . . . . 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 407	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ ∩ 𝑆 = suc 𝑧) → suc (𝐶 +o 𝑧) ⊆ 𝐷)
80	57, 79	eqsstrd 4019	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ ∩ 𝑆 = suc 𝑧) → (𝐶 +o ∩ 𝑆) ⊆ 𝐷)
81	80	rexlimdva2 3157	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐶 +o ∩ 𝑆) ⊆ 𝐷))
82		oalim 8528	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧))
83	8, 82	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧))
84		onelon 6386	. . . . . . . . . . . . . . 15 ⊢ ((∩ 𝑆 ∈ On ∧ 𝑧 ∈ ∩ 𝑆) → 𝑧 ∈ On)
85	15, 84	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ∩ 𝑆) → 𝑧 ∈ On)
86	85	ex 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ ∩ 𝑆 → 𝑧 ∈ On))
87	86	ancrd 552	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ ∩ 𝑆 → (𝑧 ∈ On ∧ 𝑧 ∈ ∩ 𝑆)))
88	74	expimpd 454	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑧 ∈ On ∧ 𝑧 ∈ ∩ 𝑆) → (𝐶 +o 𝑧) ∈ 𝐷))
89		onelss 6403	. . . . . . . . . . . . 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 3145	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧) ⊆ 𝐷)
93		iunss 5047	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧) ⊆ 𝐷 ↔ ∀𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧) ⊆ 𝐷)
94	92, 93	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧) ⊆ 𝐷)
95	94	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → ∪ 𝑧 ∈ ∩ 𝑆(𝐶 +o 𝑧) ⊆ 𝐷)
96	83, 95	eqsstrd 4019	. . . . . . 7 ⊢ ((𝜑 ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐶 +o ∩ 𝑆) ⊆ 𝐷)
97	96	ex 413	. . . . . 6 ⊢ (𝜑 → ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐶 +o ∩ 𝑆) ⊆ 𝐷))
98	53, 81, 97	3jaod 1428	. . . . 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 3612	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐷 ⊆ (𝐶 +o 𝐷)) → ∃𝑦 ∈ On 𝐷 ⊆ (𝐶 +o 𝑦))
101	5, 10, 100	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ On 𝐷 ⊆ (𝐶 +o 𝑦))
102		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑦𝐷
103		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
104		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑦 +o
105		nfrab1 3451	. . . . . . . . . 10 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}
106	105	nfint 4959	. . . . . . . . 9 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}
107	103, 104, 106	nfov 7435	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐶 +o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
108	102, 107	nfss 3973	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ (𝐶 +o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
109		oveq2 7413	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)} → (𝐶 +o 𝑦) = (𝐶 +o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}))
110	109	sseq2d 4013	. . . . . . 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 7778	. . . . . 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 7416	. . . . 5 ⊢ (𝐶 +o ∩ 𝑆) = (𝐶 +o ∩ {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)})
114	112, 113	sseqtrrdi 4032	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ (𝐶 +o ∩ 𝑆))
115	99, 114	eqssd 3998	. . 3 ⊢ (𝜑 → (𝐶 +o ∩ 𝑆) = 𝐷)
116		omelon 9637	. . . . . 6 ⊢ ω ∈ On
117		omcl 8532	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
118	116, 5, 117	sylancr 587	. . . . 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 512	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
123		ontr1 6407	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
124	119, 122, 123	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
125		nnon 7857	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
126	124, 125	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
127		oaword1 8548	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
128	118, 126, 127	syl2anc 584	. . . 4 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
129		naddwordnex.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))
130	129	oveq1d 7420	. . . . 5 ⊢ (𝜑 → (𝐴 +o (ω ·o ∩ 𝑆)) = (((ω ·o 𝐶) +o 𝑀) +o (ω ·o ∩ 𝑆)))
131		omcl 8532	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
132	116, 8, 131	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
133		nnon 7857	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On)
134	121, 133	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ On)
135		omcl 8532	. . . . . . 7 ⊢ ((ω ∈ On ∧ ∩ 𝑆 ∈ On) → (ω ·o ∩ 𝑆) ∈ On)
136	116, 15, 135	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ω ·o ∩ 𝑆) ∈ On)
137		oaass 8557	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → (((ω ·o 𝐶) +o 𝑀) +o (ω ·o ∩ 𝑆)) = ((ω ·o 𝐶) +o (𝑀 +o (ω ·o ∩ 𝑆))))
139	15, 116	jctil 520	. . . . . . . . 9 ⊢ (𝜑 → (ω ∈ On ∧ ∩ 𝑆 ∈ On))
140		omword1 8569	. . . . . . . . 9 ⊢ (((ω ∈ On ∧ ∩ 𝑆 ∈ On) ∧ ∅ ∈ ∩ 𝑆) → ω ⊆ (ω ·o ∩ 𝑆))
141	139, 40, 140	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ω ⊆ (ω ·o ∩ 𝑆))
142		oaabs 8643	. . . . . . . 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 836	. . . . . . 7 ⊢ (𝜑 → (𝑀 +o (ω ·o ∩ 𝑆)) = (ω ·o ∩ 𝑆))
144	143	oveq2d 7421	. . . . . 6 ⊢ (𝜑 → ((ω ·o 𝐶) +o (𝑀 +o (ω ·o ∩ 𝑆))) = ((ω ·o 𝐶) +o (ω ·o ∩ 𝑆)))
145		odi 8575	. . . . . . 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 1371	. . . . . 6 ⊢ (𝜑 → (ω ·o (𝐶 +o ∩ 𝑆)) = ((ω ·o 𝐶) +o (ω ·o ∩ 𝑆)))
147	115	oveq2d 7421	. . . . . 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 4028	. . 3 ⊢ (𝜑 → (𝐴 +o (ω ·o ∩ 𝑆)) ⊆ 𝐵)
152		naddcl 8672	. . . . . . . 8 ⊢ (((ω ·o 𝐶) ∈ On ∧ (ω ·o ∩ 𝑆) ∈ On) → ((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)) ∈ On)
153	132, 136, 152	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)) ∈ On)
154	118, 153, 134	3jca 1128	. . . . . 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 42130	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((ω ·o 𝐶) +o (ω ·o ∩ 𝑆)) ⊆ ((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)))
158	155, 157	eqsstrd 4019	. . . . . 6 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)))
159		oawordri 8546	. . . . . 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 42131	. . . . . . . . 9 ⊢ (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ ω) → ((ω ·o 𝐶) +o 𝑀) = ((ω ·o 𝐶) +no 𝑀))
162	132, 121, 161	syl2anc 584	. . . . . . . 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 7420	. . . . . 6 ⊢ (𝜑 → (𝐴 +no (ω ·o ∩ 𝑆)) = (((ω ·o 𝐶) +no 𝑀) +no (ω ·o ∩ 𝑆)))
165		naddass 8691	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝜑 → (((ω ·o 𝐶) +no 𝑀) +no (ω ·o ∩ 𝑆)) = ((ω ·o 𝐶) +no (𝑀 +no (ω ·o ∩ 𝑆))))
167		naddcom 8677	. . . . . . . . 9 ⊢ ((𝑀 ∈ On ∧ (ω ·o ∩ 𝑆) ∈ On) → (𝑀 +no (ω ·o ∩ 𝑆)) = ((ω ·o ∩ 𝑆) +no 𝑀))
168	134, 136, 167	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀 +no (ω ·o ∩ 𝑆)) = ((ω ·o ∩ 𝑆) +no 𝑀))
169	168	oveq2d 7421	. . . . . . 7 ⊢ (𝜑 → ((ω ·o 𝐶) +no (𝑀 +no (ω ·o ∩ 𝑆))) = ((ω ·o 𝐶) +no ((ω ·o ∩ 𝑆) +no 𝑀)))
170		naddonnn 42131	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → (((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)) +o 𝑀) = (((ω ·o 𝐶) +no (ω ·o ∩ 𝑆)) +no 𝑀))
172		naddass 8691	. . . . . . . . 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 1371	. . . . . . . 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 4021	. . . 4 ⊢ (𝜑 → ((ω ·o 𝐷) +o 𝑀) ⊆ (𝐴 +no (ω ·o ∩ 𝑆)))
178	134, 118	jca 512	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ On ∧ (ω ·o 𝐷) ∈ On))
179		oaordi 8542	. . . . . 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 2833	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ((ω ·o 𝐷) +o 𝑀))
182	177, 181	sseldd 3982	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 +no (ω ·o ∩ 𝑆)))
183	115, 151, 182	3jca 1128	. 2 ⊢ (𝜑 → ((𝐶 +o ∩ 𝑆) = 𝐷 ∧ (𝐴 +o (ω ·o ∩ 𝑆)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o ∩ 𝑆))))
184		oveq2 7413	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐶 +o 𝑥) = (𝐶 +o ∩ 𝑆))
185	184	eqeq1d 2734	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → ((𝐶 +o 𝑥) = 𝐷 ↔ (𝐶 +o ∩ 𝑆) = 𝐷))
186		oveq2 7413	. . . . . 6 ⊢ (𝑥 = ∩ 𝑆 → (ω ·o 𝑥) = (ω ·o ∩ 𝑆))
187	186	oveq2d 7421	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +o (ω ·o 𝑥)) = (𝐴 +o (ω ·o ∩ 𝑆)))
188	187	sseq1d 4012	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → ((𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ↔ (𝐴 +o (ω ·o ∩ 𝑆)) ⊆ 𝐵))
189	186	oveq2d 7421	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +no (ω ·o 𝑥)) = (𝐴 +no (ω ·o ∩ 𝑆)))
190	189	eleq2d 2819	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → (𝐵 ∈ (𝐴 +no (ω ·o 𝑥)) ↔ 𝐵 ∈ (𝐴 +no (ω ·o ∩ 𝑆))))
191	185, 188, 190	3anbi123d 1436	. . 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 3612	. 2 ⊢ ((∩ 𝑆 ∈ (On ∖ 1o) ∧ ((𝐶 +o ∩ 𝑆) = 𝐷 ∧ (𝐴 +o (ω ·o ∩ 𝑆)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o ∩ 𝑆)))) → ∃𝑥 ∈ (On ∖ 1o)((𝐶 +o 𝑥) = 𝐷 ∧ (𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o 𝑥))))
193	42, 183, 192	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1o)((𝐶 +o 𝑥) = 𝐷 ∧ (𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321  ∩ cint 4949  ∪ ciun 4996  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7405  ωcom 7851  1_oc1o 8455   +_o coa 8459   ·_o comu 8460   +no cnadd 8660

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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632

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-rmo 3376  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-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  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-se 5631  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-pred 6297  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-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-nadd 8661

This theorem is referenced by: (None)