Theorem omeulem1 7816

Description: Lemma for omeu 7819: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
omeulem1	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)

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

Proof of Theorem omeulem1

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1131	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ On)
2		sucelon 7164	. . . . . 6 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 208	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ∈ On)
4		simp1 1130	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
5		on0eln0 5923	. . . . . . 7 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
6	5	biimpar 463	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
7	6	3adant2 1125	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
8		omword2 7808	. . . . 5 ⊢ (((suc 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵))
9	3, 4, 7, 8	syl21anc 1475	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵))
10		sucidg 5946	. . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
11		ssel 3746	. . . . 5 ⊢ (suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵) → (𝐵 ∈ suc 𝐵 → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
12	10, 11	syl5 34	. . . 4 ⊢ (suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵) → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
13	9, 1, 12	sylc 65	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵))
14		suceq 5933	. . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	oveq2d 6809	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝐵))
16	15	eleq2d 2836	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
17	16	rspcev 3460	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
18	1, 13, 17	syl2anc 573	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
19		suceq 5933	. . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
20	19	oveq2d 6809	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝑧))
21	20	eleq2d 2836	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
22	21	onminex 7154	. . 3 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
23		vex 3354	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
24	23	elon 5875	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
25		ordzsl 7192	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥))
26	24, 25	bitri 264	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥))
27		noel 4067	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝐵 ∈ ∅
28		oveq2 6801	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
29		om0x 7753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ·𝑜 ∅) = ∅
30	28, 29	syl6eq 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = ∅)
31	30	eleq2d 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∅))
32	27, 31	mtbiri 316	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
33	32	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
34		simp3 1132	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → 𝑥 = suc 𝑤)
35		simp2 1131	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))
36		raleq 3287	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = suc 𝑤 → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
37		vex 3354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑤 ∈ V
38	37	sucid 5947	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ suc 𝑤
39		suceq 5933	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤)
40	39	oveq2d 6809	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑤 → (𝐴 ·𝑜 suc 𝑧) = (𝐴 ·𝑜 suc 𝑤))
41	40	eleq2d 2836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
42	41	notbid 307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
43	42	rspcv 3456	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ suc 𝑤 → (∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
44	38, 43	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))
45	36, 44	syl6bi 243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = suc 𝑤 → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
46	34, 35, 45	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))
47		oveq2 6801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = suc 𝑤 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑤))
48	47	eleq2d 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = suc 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
49	48	notbid 307	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = suc 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
50	49	biimpar 463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = suc 𝑤 ∧ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
51	34, 46, 50	syl2anc 573	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
52	51	3expia 1114	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
53	52	rexlimdvw 3182	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (∃𝑤 ∈ On 𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
54		ralnex 3141	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))
55		simpr 471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
56	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 𝑥 ∈ V)
57		simpl 468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → Lim 𝑥)
58		omlim 7767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧))
59	55, 56, 57, 58	syl12anc 1474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧))
60	59	eleq2d 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧)))
61		eliun 4658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ↔ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧))
62		limord 5927	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Lim 𝑥 → Ord 𝑥)
63	62	3ad2ant1 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → Ord 𝑥)
64	63, 24	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ On)
65		simp3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
66		onelon 5891	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
67	64, 65, 66	syl2anc 573	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
68		suceloni 7160	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ On → suc 𝑧 ∈ On)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → suc 𝑧 ∈ On)
70		simp2 1131	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝐴 ∈ On)
71		sssucid 5945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑧 ⊆ suc 𝑧
72		omwordi 7805	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ On ∧ suc 𝑧 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ suc 𝑧 → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧)))
73	71, 72	mpi 20	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ On ∧ suc 𝑧 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))
74	67, 69, 70, 73	syl3anc 1476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))
75	74	sseld 3751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
76	75	3expia 1114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝑧 ∈ 𝑥 → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))))
77	76	reximdvai 3163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
78	61, 77	syl5bi 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
79	60, 78	sylbid 230	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
80	79	con3d 149	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (¬ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
81	54, 80	syl5bi 232	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
82	81	expimpd 441	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
83	82	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
84	83	3ad2antl1 1200	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
85	33, 53, 84	3jaod 1540	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ((𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
86	26, 85	syl5bi 232	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
87	86	impr 442	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
88		simpl1 1227	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐴 ∈ On)
89		simprr 756	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝑥 ∈ On)
90		omcl 7770	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
91	88, 89, 90	syl2anc 573	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ∈ On)
92		simpl2 1229	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐵 ∈ On)
93		ontri1 5900	. . . . . . . . . . . 12 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
94	91, 92, 93	syl2anc 573	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
95	87, 94	mpbird 247	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ⊆ 𝐵)
96		oawordex 7791	. . . . . . . . . . 11 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
97	91, 92, 96	syl2anc 573	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
98	95, 97	mpbid 222	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
99	98	3adantr1 1174	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
100		simp3r 1244	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
101		simp21 1248	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
102		simp11 1245	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐴 ∈ On)
103		simp23 1250	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑥 ∈ On)
104		omsuc 7760	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
105	102, 103, 104	syl2anc 573	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
106	101, 105	eleqtrd 2852	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐵 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
107	100, 106	eqeltrd 2850	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
108		simp3l 1243	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦 ∈ On)
109	102, 103, 90	syl2anc 573	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝐴 ·𝑜 𝑥) ∈ On)
110		oaord 7781	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
111	108, 102, 109, 110	syl3anc 1476	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
112	107, 111	mpbird 247	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦 ∈ 𝐴)
113	112, 100	jca 501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦 ∈ 𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
114	113	3expia 1114	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) → (𝑦 ∈ 𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
115	114	reximdv2 3162	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵 → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
116	99, 115	mpd 15	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
117	116	expcom 398	. . . . . 6 ⊢ ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
118	117	3expia 1114	. . . . 5 ⊢ ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
119	118	com13 88	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (𝑥 ∈ On → ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
120	119	reximdvai 3163	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
121	22, 120	syl5 34	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
122	18, 121	mpd 15	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ w3o 1070   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ⊆ wss 3723  ∅c0 4063  ∪ ciun 4654  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793   +_𝑜 coa 7710   ·_𝑜 comu 7711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718

This theorem is referenced by:  omeu  7819