MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnawordex Structured version   Visualization version   GIF version

Theorem nnawordex 7581
Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnawordex ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simplr 787 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
2 nnon 6940 . . . . . . . 8 (𝐵 ∈ ω → 𝐵 ∈ On)
31, 2syl 17 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ On)
4 simpll 785 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
5 nnaword2 7574 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
61, 4, 5syl2anc 690 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
7 oveq2 6535 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
87sseq2d 3595 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
98elrab 3330 . . . . . . 7 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
103, 6, 9sylanbrc 694 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
11 intss1 4421 . . . . . 6 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵)
1210, 11syl 17 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵)
13 ssrab2 3649 . . . . . . . 8 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
14 ne0i 3879 . . . . . . . . 9 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅)
1510, 14syl 17 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅)
16 oninton 6869 . . . . . . . 8 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
1713, 15, 16sylancr 693 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
18 eloni 5636 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On → Ord {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
1917, 18syl 17 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
20 ordom 6943 . . . . . 6 Ord ω
21 ordtr2 5671 . . . . . 6 ((Ord {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∧ Ord ω) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
2219, 20, 21sylancl 692 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
2312, 1, 22mp2and 710 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω)
24 nna0 7548 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
2524ad2antrr 757 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
26 simpr 475 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴𝐵)
2725, 26eqsstrd 3601 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ⊆ 𝐵)
28 oveq2 6535 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 ∅))
2928sseq1d 3594 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → ((𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +𝑜 ∅) ⊆ 𝐵))
3027, 29syl5ibrcom 235 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵))
31 simprr 791 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
3231oveq2d 6543 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 suc 𝑥))
334adantr 479 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
34 simprl 789 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
35 nnasuc 7550 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3633, 34, 35syl2anc 690 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3732, 36eqtrd 2643 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = suc (𝐴 +𝑜 𝑥))
38 nnord 6942 . . . . . . . . . . 11 (𝐵 ∈ ω → Ord 𝐵)
391, 38syl 17 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord 𝐵)
4039adantr 479 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord 𝐵)
41 nnon 6940 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
4241adantr 479 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
43 vex 3175 . . . . . . . . . . . . . 14 𝑥 ∈ V
4443sucid 5707 . . . . . . . . . . . . 13 𝑥 ∈ suc 𝑥
45 simpr 475 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
4644, 45syl5eleqr 2694 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
47 oveq2 6535 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑥))
4847sseq2d 3595 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
4948onnminsb 6873 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
5042, 46, 49sylc 62 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
5150adantl 480 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
52 nnacl 7555 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
5333, 34, 52syl2anc 690 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ ω)
54 nnord 6942 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
5553, 54syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord (𝐴 +𝑜 𝑥))
56 ordtri1 5659 . . . . . . . . . . . 12 ((Ord 𝐵 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
5740, 55, 56syl2anc 690 . . . . . . . . . . 11 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
5857con2bid 342 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ((𝐴 +𝑜 𝑥) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
5951, 58mpbird 245 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ 𝐵)
60 ordsucss 6887 . . . . . . . . 9 (Ord 𝐵 → ((𝐴 +𝑜 𝑥) ∈ 𝐵 → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵))
6140, 59, 60sylc 62 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵)
6237, 61eqsstrd 3601 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
6362rexlimdvaa 3013 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (∃𝑥 ∈ ω {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥 → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵))
64 nn0suc 6959 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥))
6523, 64syl 17 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥))
6630, 63, 65mpjaod 394 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
67 onint 6864 . . . . . . 7 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
6813, 15, 67sylancr 693 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
69 nfrab1 3098 . . . . . . . . 9 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7069nfint 4415 . . . . . . . 8 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
71 nfcv 2750 . . . . . . . 8 𝑦On
72 nfcv 2750 . . . . . . . . 9 𝑦𝐵
73 nfcv 2750 . . . . . . . . . 10 𝑦𝐴
74 nfcv 2750 . . . . . . . . . 10 𝑦 +𝑜
7573, 74, 70nfov 6553 . . . . . . . . 9 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7672, 75nfss 3560 . . . . . . . 8 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
77 oveq2 6535 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7877sseq2d 3595 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
7970, 71, 76, 78elrabf 3328 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
8079simprbi 478 . . . . . 6 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
8168, 80syl 17 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
8266, 81eqssd 3584 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵)
83 oveq2 6535 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
8483eqeq1d 2611 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵))
8584rspcev 3281 . . . 4 (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω ∧ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
8623, 82, 85syl2anc 690 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
8786ex 448 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
88 nnaword1 7573 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
8988adantlr 746 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
90 sseq2 3589 . . . 4 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
9189, 90syl5ibcom 233 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
9291rexlimdva 3012 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
9387, 92impbid 200 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  {crab 2899  wss 3539  c0 3873   cint 4404  Ord word 5625  Oncon0 5626  suc csuc 5628  (class class class)co 6527  ωcom 6934   +𝑜 coa 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428
This theorem is referenced by:  nnaordex  7582  unfilem1  8086  hashdom  12981
  Copyright terms: Public domain W3C validator