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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 ∃𝑦𝐴 ((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.
StepHypRef Expression
1 simp2 1160 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ On)
2 sucelon 7241 . . . . . 6 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
31, 2sylib 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 → (∅ ∈ 𝐴𝐴 ≠ ∅))
65biimpar 465 . . . . . 6 ((𝐴 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
763adant2 1154 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
8 omword2 7885 . . . . 5 (((suc 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵))
93, 4, 7, 8syl21anc 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 𝐵𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
1210, 11syl5 34 . . . 4 (suc 𝐵 ⊆ (𝐴 ·𝑜 suc 𝐵) → (𝐵 ∈ On → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
139, 1, 12sylc 65 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵))
14 suceq 5997 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
1514oveq2d 6884 . . . . 5 (𝑥 = 𝐵 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝐵))
1615eleq2d 2867 . . . 4 (𝑥 = 𝐵 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)))
1716rspcev 3498 . . 3 ((𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ·𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
181, 13, 17syl2anc 575 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥))
19 suceq 5997 . . . . . 6 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2019oveq2d 6884 . . . . 5 (𝑥 = 𝑧 → (𝐴 ·𝑜 suc 𝑥) = (𝐴 ·𝑜 suc 𝑧))
2120eleq2d 2867 . . . 4 (𝑥 = 𝑧 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
2221onminex 7231 . . 3 (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
23 vex 3390 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2423elon 5939 . . . . . . . . . . . . . 14 (𝑥 ∈ On ↔ Ord 𝑥)
25 ordzsl 7269 . . . . . . . . . . . . . 14 (Ord 𝑥 ↔ (𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥))
2624, 25bitri 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 ∅) = ∅
3028, 29syl6eq 2852 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = ∅)
3130eleq2d 2867 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ ∅))
3227, 31mtbiri 318 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
3332a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = ∅ → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
34 simp3 1161 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → 𝑥 = suc 𝑤)
35 simp2 1160 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))
36 raleq 3323 . . . . . . . . . . . . . . . . . . 19 (𝑥 = suc 𝑤 → (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
37 vex 3390 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
3837sucid 6011 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ suc 𝑤
39 suceq 5997 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤)
4039oveq2d 6884 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑤 → (𝐴 ·𝑜 suc 𝑧) = (𝐴 ·𝑜 suc 𝑤))
4140eleq2d 2867 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
4241notbid 309 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
4342rspcv 3494 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ suc 𝑤 → (∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
4438, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (∀𝑧 ∈ suc 𝑤 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤))
4536, 44syl6bi 244 . . . . . . . . . . . . . . . . . 18 (𝑥 = suc 𝑤 → (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
4634, 35, 45sylc 65 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 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))
4847eleq2d 2867 . . . . . . . . . . . . . . . . . . 19 (𝑥 = suc 𝑤 → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
4948notbid 309 . . . . . . . . . . . . . . . . . 18 (𝑥 = suc 𝑤 → (¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)))
5049biimpar 465 . . . . . . . . . . . . . . . . 17 ((𝑥 = suc 𝑤 ∧ ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑤)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
5134, 46, 50syl2anc 575 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 = suc 𝑤) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
52513expia 1143 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
5352rexlimdvw 3218 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (∃𝑤 ∈ On 𝑥 = suc 𝑤 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
54 ralnex 3176 . . . . . . . . . . . . . . . . . 18 (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ↔ ¬ ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))
55 simpr 473 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥𝐴 ∈ On) → 𝐴 ∈ On)
5623a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥𝐴 ∈ On) → 𝑥 ∈ V)
57 simpl 470 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥𝐴 ∈ On) → Lim 𝑥)
58 omlim 7844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = 𝑧𝑥 (𝐴 ·𝑜 𝑧))
5955, 56, 57, 58syl12anc 856 . . . . . . . . . . . . . . . . . . . . 21 ((Lim 𝑥𝐴 ∈ On) → (𝐴 ·𝑜 𝑥) = 𝑧𝑥 (𝐴 ·𝑜 𝑧))
6059eleq2d 2867 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) ↔ 𝐵 𝑧𝑥 (𝐴 ·𝑜 𝑧)))
61 eliun 4709 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 𝑧𝑥 (𝐴 ·𝑜 𝑧) ↔ ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧))
62 limord 5991 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑥 → Ord 𝑥)
63623ad2ant1 1156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → Ord 𝑥)
6463, 24sylibr 225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → 𝑥 ∈ On)
65 simp3 1161 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → 𝑧𝑥)
66 onelon 5955 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
6764, 65, 66syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
68 suceloni 7237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ On → suc 𝑧 ∈ On)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → suc 𝑧 ∈ On)
70 simp2 1160 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → 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)))
7371, 72mpi 20 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 ∈ On ∧ suc 𝑧 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))
7467, 69, 70, 73syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 suc 𝑧))
7574sseld 3791 . . . . . . . . . . . . . . . . . . . . . . 23 ((Lim 𝑥𝐴 ∈ On ∧ 𝑧𝑥) → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
76753expia 1143 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥𝐴 ∈ On) → (𝑧𝑥 → (𝐵 ∈ (𝐴 ·𝑜 𝑧) → 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧))))
7776reximdvai 3198 . . . . . . . . . . . . . . . . . . . . 21 ((Lim 𝑥𝐴 ∈ On) → (∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 𝑧) → ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
7861, 77syl5bi 233 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐴 ∈ On) → (𝐵 𝑧𝑥 (𝐴 ·𝑜 𝑧) → ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
7960, 78sylbid 231 . . . . . . . . . . . . . . . . . . 19 ((Lim 𝑥𝐴 ∈ On) → (𝐵 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)))
8079con3d 149 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴 ∈ On) → (¬ ∃𝑧𝑥 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8154, 80syl5bi 233 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐴 ∈ On) → (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8281expimpd 443 . . . . . . . . . . . . . . . 16 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8382com12 32 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
84833ad2antl1 1229 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (Lim 𝑥 → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8533, 53, 843jaod 1546 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ((𝑥 = ∅ ∨ ∃𝑤 ∈ On 𝑥 = suc 𝑤 ∨ Lim 𝑥) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8626, 85syl5bi 233 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
8786impr 444 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥))
88 simpl1 1235 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝐴 ∈ On)
89 simprr 780 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → 𝑥 ∈ On)
90 omcl 7847 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
9188, 89, 90syl2anc 575 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (𝐴 ·𝑜 𝑥) ∈ On)
92 simpl2 1237 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 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)))
9491, 92, 93syl2anc 575 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ·𝑜 𝑥)))
9587, 94mpbird 248 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 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))
9791, 92, 96syl2anc 575 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝑥) ⊆ 𝐵 ↔ ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
9895, 97mpbid 223 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
99983adantr1 1203 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 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) ∧ ∀𝑧𝑥 ¬ 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) ∧ ∀𝑧𝑥 ¬ 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) ∧ ∀𝑧𝑥 ¬ 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) ∧ ∀𝑧𝑥 ¬ 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))
105102, 103, 104syl2anc 575 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
106101, 105eleqtrd 2883 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝐵 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
107100, 106eqeltrd 2881 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 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) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦 ∈ On)
109102, 103, 90syl2anc 575 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 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) → (𝑦𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
111108, 102, 109, 110syl3anc 1483 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
112107, 111mpbird 248 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → 𝑦𝐴)
113112, 100jca 503 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) ∧ (𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)) → (𝑦𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
1141133expia 1143 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ((𝑦 ∈ On ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵) → (𝑦𝐴 ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
115114reximdv2 3197 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → (∃𝑦 ∈ On ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵 → ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
11699, 115mpd 15 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On)) → ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
117116expcom 400 . . . . . 6 ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧) ∧ 𝑥 ∈ On) → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
1181173expia 1143 . . . . 5 ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → (𝑥 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
119118com13 88 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (𝑥 ∈ On → ((𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)))
120119reximdvai 3198 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On (𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) ∧ ∀𝑧𝑥 ¬ 𝐵 ∈ (𝐴 ·𝑜 suc 𝑧)) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
12122, 120syl5 34 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ·𝑜 suc 𝑥) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))
12218, 121mpd 15 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
Colors of variables: wff setvar class
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
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