Theorem oawordeulem 8174

Description: Lemma for oawordex 8177. (Contributed by NM, 11-Dec-2004.)

Hypotheses

Ref	Expression
oawordeulem.1	⊢ 𝐴 ∈ On
oawordeulem.2	⊢ 𝐵 ∈ On
oawordeulem.3	⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}

Assertion

Ref	Expression
oawordeulem	⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem oawordeulem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oawordeulem.3	. . . . 5 ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
2	1	ssrab3 4056	. . . 4 ⊢ 𝑆 ⊆ On
3		oawordeulem.2	. . . . . 6 ⊢ 𝐵 ∈ On
4		oawordeulem.1	. . . . . . 7 ⊢ 𝐴 ∈ On
5		oaword2 8173	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵))
6	3, 4, 5	mp2an 690	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 +o 𝐵)
7		oveq2 7158	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
8	7	sseq2d 3998	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
9	8, 1	elrab2 3682	. . . . . 6 ⊢ (𝐵 ∈ 𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)))
10	3, 6, 9	mpbir2an 709	. . . . 5 ⊢ 𝐵 ∈ 𝑆
11	10	ne0ii 4302	. . . 4 ⊢ 𝑆 ≠ ∅
12		oninton 7509	. . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ On)
13	2, 11, 12	mp2an 690	. . 3 ⊢ ∩ 𝑆 ∈ On
14		onzsl 7555	. . . . . 6 ⊢ (∩ 𝑆 ∈ On ↔ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
15	13, 14	mpbi 232	. . . . 5 ⊢ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆))
16		oveq2 7158	. . . . . . . . 9 ⊢ (∩ 𝑆 = ∅ → (𝐴 +o ∩ 𝑆) = (𝐴 +o ∅))
17		oa0 8135	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18	4, 17	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 +o ∅) = 𝐴
19	16, 18	syl6eq 2872	. . . . . . . 8 ⊢ (∩ 𝑆 = ∅ → (𝐴 +o ∩ 𝑆) = 𝐴)
20	19	sseq1d 3997	. . . . . . 7 ⊢ (∩ 𝑆 = ∅ → ((𝐴 +o ∩ 𝑆) ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
21	20	biimprd 250	. . . . . 6 ⊢ (∩ 𝑆 = ∅ → (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵))
22		oveq2 7158	. . . . . . . . . 10 ⊢ (∩ 𝑆 = suc 𝑧 → (𝐴 +o ∩ 𝑆) = (𝐴 +o suc 𝑧))
23		oasuc 8143	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
24	4, 23	mpan 688	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧))
25	22, 24	sylan9eqr 2878	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → (𝐴 +o ∩ 𝑆) = suc (𝐴 +o 𝑧))
26		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27	26	sucid 6264	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ suc 𝑧
28		eleq2 2901	. . . . . . . . . . . 12 ⊢ (∩ 𝑆 = suc 𝑧 → (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ suc 𝑧))
29	27, 28	mpbiri 260	. . . . . . . . . . 11 ⊢ (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆)
30	13	oneli 6292	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∩ 𝑆 → 𝑧 ∈ On)
31	1	inteqi 4872	. . . . . . . . . . . . . . 15 ⊢ ∩ 𝑆 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
32	31	eleq2i 2904	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
33		oveq2 7158	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝐴 +o 𝑦) = (𝐴 +o 𝑧))
34	33	sseq2d 3998	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑧)))
35	34	onnminsb 7513	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
36	32, 35	syl5bi 244	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ 𝑆 → ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
37		oacl 8154	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +o 𝑧) ∈ On)
38	4, 37	mpan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝐴 +o 𝑧) ∈ On)
39		ontri1 6219	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ (𝐴 +o 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
40	3, 38, 39	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +o 𝑧) ↔ ¬ (𝐴 +o 𝑧) ∈ 𝐵))
41	40	con2bid 357	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ On → ((𝐴 +o 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +o 𝑧)))
42	36, 41	sylibrd 261	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵))
43	30, 42	mpcom 38	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∩ 𝑆 → (𝐴 +o 𝑧) ∈ 𝐵)
44	3	onordi 6289	. . . . . . . . . . . 12 ⊢ Ord 𝐵
45		ordsucss 7527	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → ((𝐴 +o 𝑧) ∈ 𝐵 → suc (𝐴 +o 𝑧) ⊆ 𝐵))
46	44, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑧) ∈ 𝐵 → suc (𝐴 +o 𝑧) ⊆ 𝐵)
47	29, 43, 46	3syl 18	. . . . . . . . . 10 ⊢ (∩ 𝑆 = suc 𝑧 → suc (𝐴 +o 𝑧) ⊆ 𝐵)
48	47	adantl 484	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → suc (𝐴 +o 𝑧) ⊆ 𝐵)
49	25, 48	eqsstrd 4004	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → (𝐴 +o ∩ 𝑆) ⊆ 𝐵)
50	49	rexlimiva 3281	. . . . . . 7 ⊢ (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵)
51	50	a1d 25	. . . . . 6 ⊢ (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵))
52		oalim 8151	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐴 +o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +o 𝑧))
53	4, 52	mpan 688	. . . . . . . 8 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 +o ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +o 𝑧))
54		iunss 4961	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ ∩ 𝑆(𝐴 +o 𝑧) ⊆ 𝐵 ↔ ∀𝑧 ∈ ∩ 𝑆(𝐴 +o 𝑧) ⊆ 𝐵)
55	3	onelssi 6293	. . . . . . . . . 10 ⊢ ((𝐴 +o 𝑧) ∈ 𝐵 → (𝐴 +o 𝑧) ⊆ 𝐵)
56	43, 55	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ ∩ 𝑆 → (𝐴 +o 𝑧) ⊆ 𝐵)
57	54, 56	mprgbir 3153	. . . . . . . 8 ⊢ ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +o 𝑧) ⊆ 𝐵
58	53, 57	eqsstrdi 4020	. . . . . . 7 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 +o ∩ 𝑆) ⊆ 𝐵)
59	58	a1d 25	. . . . . 6 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵))
60	21, 51, 59	3jaoi 1423	. . . . 5 ⊢ ((∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵))
61	15, 60	ax-mp 5	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) ⊆ 𝐵)
62	8	rspcev 3622	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +o 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦))
63	3, 6, 62	mp2an 690	. . . . . 6 ⊢ ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦)
64		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
65		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
66		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑦 +o
67		nfrab1 3384	. . . . . . . . . 10 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
68	67	nfint 4878	. . . . . . . . 9 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
69	65, 66, 68	nfov 7180	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
70	64, 69	nfss 3959	. . . . . . 7 ⊢ Ⅎ𝑦 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
71		oveq2 7158	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
72	71	sseq2d 3998	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
73	70, 72	onminsb 7508	. . . . . 6 ⊢ (∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +o 𝑦) → 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
74	63, 73	ax-mp 5	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
75	31	oveq2i 7161	. . . . 5 ⊢ (𝐴 +o ∩ 𝑆) = (𝐴 +o ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
76	74, 75	sseqtrri 4003	. . . 4 ⊢ 𝐵 ⊆ (𝐴 +o ∩ 𝑆)
77		eqss 3981	. . . 4 ⊢ ((𝐴 +o ∩ 𝑆) = 𝐵 ↔ ((𝐴 +o ∩ 𝑆) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 +o ∩ 𝑆)))
78	61, 76, 77	sylanblrc 592	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 +o ∩ 𝑆) = 𝐵)
79		oveq2 7158	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +o 𝑥) = (𝐴 +o ∩ 𝑆))
80	79	eqeq1d 2823	. . . 4 ⊢ (𝑥 = ∩ 𝑆 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∩ 𝑆) = 𝐵))
81	80	rspcev 3622	. . 3 ⊢ ((∩ 𝑆 ∈ On ∧ (𝐴 +o ∩ 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
82	13, 78, 81	sylancr 589	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
83		eqtr3 2843	. . . 4 ⊢ (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
84		oacan 8168	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
85	4, 84	mp3an1 1444	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑥) = (𝐴 +o 𝑦) ↔ 𝑥 = 𝑦))
86	83, 85	syl5ib 246	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦))
87	86	rgen2 3203	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)
88		oveq2 7158	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
89	88	eqeq1d 2823	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o 𝑦) = 𝐵))
90	89	reu4 3721	. 2 ⊢ (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ↔ (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +o 𝑥) = 𝐵 ∧ (𝐴 +o 𝑦) = 𝐵) → 𝑥 = 𝑦)))
91	82, 87, 90	sylanblrc 592	1 ⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ w3o 1082   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  ∩ cint 4868  ∪ ciun 4911  Ord word 6184  Oncon0 6185  Lim wlim 6186  suc csuc 6187  (class class class)co 7150   +_o coa 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100

This theorem is referenced by:  oawordeu  8175