Theorem omeulem1 7893

Description: Lemma for omeu 7896: 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 1160	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ On)
2		sucelon 7241	. . . . . 6 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 209	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ∈ On)
4		simp1 1159	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
5		on0eln0 5987	. . . . . . 7 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
6	5	biimpar 465	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
7	6	3adant2 1154	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
8		omword2 7885	. . . . 5 ⊢ (((suc 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵))
9	3, 4, 7, 8	syl21anc 857	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵))
10		sucidg 6010	. . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
11		ssel 3786	. . . . 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 5997	. . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	oveq2d 6884	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝐵))
16	15	eleq2d 2867	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
17	16	rspcev 3498	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
18	1, 13, 17	syl2anc 575	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
19		suceq 5997	. . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
20	19	oveq2d 6884	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝑧))
21	20	eleq2d 2867	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
22	21	onminex 7231	. . 3 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
23		vex 3390	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
24	23	elon 5939	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
25		ordzsl 7269	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥))
26	24, 25	bitri 266	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥))
27		noel 4114	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝐵 ∈ ∅
28		oveq2 6876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
29		om0x 7830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ·𝑜 ∅) = ∅
30	28, 29	syl6eq 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = ∅)
31	30	eleq2d 2867	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∅))
32	27, 31	mtbiri 318	. . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → 𝑥 = suc 𝑤)
35		simp2 1160	. . . . . . . . . . . . . . . . . 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 3323	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = suc 𝑤 → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
37		vex 3390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑤 ∈ V
38	37	sucid 6011	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ suc 𝑤
39		suceq 5997	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤)
40	39	oveq2d 6884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑤 → (𝐴 ·𝑜 suc 𝑧) = (𝐴 ·𝑜 suc 𝑤))
41	40	eleq2d 2867	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
42	41	notbid 309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
43	42	rspcv 3494	. . . . . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = suc 𝑤 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑤))
48	47	eleq2d 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = suc 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
49	48	notbid 309	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = suc 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
50	49	biimpar 465	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = suc 𝑤 ∧ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
51	34, 46, 50	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
52	51	3expia 1143	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
53	52	rexlimdvw 3218	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (∃𝑤 ∈ On 𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
54		ralnex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))
55		simpr 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
56	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 𝑥 ∈ V)
57		simpl 470	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → Lim 𝑥)
58		omlim 7844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧))
59	55, 56, 57, 58	syl12anc 856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑥) = ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧))
60	59	eleq2d 2867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧)))
61		eliun 4709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ↔ ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧))
62		limord 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Lim 𝑥 → Ord 𝑥)
63	62	3ad2ant1 1156	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → Ord 𝑥)
64	63, 24	sylibr 225	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ On)
65		simp3 1161	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
66		onelon 5955	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
67	64, 65, 66	syl2anc 575	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
68		suceloni 7237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ On → suc 𝑧 ∈ On)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → suc 𝑧 ∈ On)
70		simp2 1160	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝐴 ∈ On)
71		sssucid 6009	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑧 ⊆ suc 𝑧
72		omwordi 7882	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))
75	74	sseld 3791	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On ∧ 𝑧 ∈ 𝑥) → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
76	75	3expia 1143	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝑧 ∈ 𝑥 → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))))
77	76	reximdvai 3198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
78	61, 77	syl5bi 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 ∈ ∪ 𝑧 ∈ 𝑥 (𝐴 ·𝑜 𝑧) → ∃𝑧 ∈ 𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
79	60, 78	sylbid 231	. . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
82	81	expimpd 443	. . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
85	33, 53, 84	3jaod 1546	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
87	86	impr 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
88		simpl1 1235	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐴 ∈ On)
89		simprr 780	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝑥 ∈ On)
90		omcl 7847	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
91	88, 89, 90	syl2anc 575	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ∈ On)
92		simpl2 1237	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐵 ∈ On)
93		ontri1 5964	. . . . . . . . . . . 12 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
94	91, 92, 93	syl2anc 575	. . . . . . . . . . 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 248	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧 ∈ 𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ⊆ 𝐵)
96		oawordex 7868	. . . . . . . . . . 11 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
97	91, 92, 96	syl2anc 575	. . . . . . . . . 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 223	. . . . . . . . 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 1203	. . . . . . . 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 1252	. . . . . . . . . . . . 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 1256	. . . . . . . . . . . . . 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 1253	. . . . . . . . . . . . . . 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 1258	. . . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
105	102, 103, 104	syl2anc 575	. . . . . . . . . . . . . 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 2883	. . . . . . . . . . . . 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 2881	. . . . . . . . . . . 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 1251	. . . . . . . . . . . . 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 575	. . . . . . . . . . . . 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 7858	. . . . . . . . . . . . 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 1483	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 503	. . . . . . . . . 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 1143	. . . . . . . . 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 3197	. . . . . . . 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 400	. . . . . 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 1143	. . . . 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 3198	. . 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 197   ∧ wa 384   ∨ w3o 1099   ∧ w3a 1100   = wceq 1637   ∈ wcel 2155   ≠ wne 2974  ∀wral 3092  ∃wrex 3093  Vcvv 3387   ⊆ wss 3763  ∅c0 4110  ∪ ciun 4705  Ord word 5929  Oncon0 5930  Lim wlim 5931  suc csuc 5932  (class class class)co 6868   +_𝑜 coa 7787   ·_𝑜 comu 7788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-omul 7795

This theorem is referenced by:  omeu  7896