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

Theorem nnawordex 7702
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 791 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
2 nnon 7056 . . . . . . . 8 (𝐵 ∈ ω → 𝐵 ∈ On)
31, 2syl 17 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ On)
4 simpll 789 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
5 nnaword2 7695 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
61, 4, 5syl2anc 692 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
7 oveq2 6643 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
87sseq2d 3625 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
98elrab 3357 . . . . . . 7 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
103, 6, 9sylanbrc 697 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
11 intss1 4483 . . . . . 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 3679 . . . . . . . 8 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
14 ne0i 3913 . . . . . . . . 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 6985 . . . . . . . 8 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
1713, 15, 16sylancr 694 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
18 eloni 5721 . . . . . . 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 7059 . . . . . 6 Ord ω
21 ordtr2 5756 . . . . . 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 693 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
2312, 1, 22mp2and 714 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω)
24 nna0 7669 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
2524ad2antrr 761 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
26 simpr 477 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴𝐵)
2725, 26eqsstrd 3631 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ⊆ 𝐵)
28 oveq2 6643 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 ∅))
2928sseq1d 3624 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → ((𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +𝑜 ∅) ⊆ 𝐵))
3027, 29syl5ibrcom 237 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵))
31 simprr 795 . . . . . . . . . 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 6651 . . . . . . . . 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 481 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
34 simprl 793 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
35 nnasuc 7671 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3633, 34, 35syl2anc 692 . . . . . . . . 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 2654 . . . . . . . 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 7058 . . . . . . . . . . 11 (𝐵 ∈ ω → Ord 𝐵)
391, 38syl 17 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord 𝐵)
4039adantr 481 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord 𝐵)
41 nnon 7056 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
4241adantr 481 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
43 vex 3198 . . . . . . . . . . . . . 14 𝑥 ∈ V
4443sucid 5792 . . . . . . . . . . . . 13 𝑥 ∈ suc 𝑥
45 simpr 477 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
4644, 45syl5eleqr 2706 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
47 oveq2 6643 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑥))
4847sseq2d 3625 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
4948onnminsb 6989 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
5042, 46, 49sylc 65 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
5150adantl 482 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
52 nnacl 7676 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
5333, 34, 52syl2anc 692 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ ω)
54 nnord 7058 . . . . . . . . . . . . 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 5744 . . . . . . . . . . . 12 ((Ord 𝐵 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
5740, 55, 56syl2anc 692 . . . . . . . . . . 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 344 . . . . . . . . . 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 247 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ 𝐵)
60 ordsucss 7003 . . . . . . . . 9 (Ord 𝐵 → ((𝐴 +𝑜 𝑥) ∈ 𝐵 → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵))
6140, 59, 60sylc 65 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵)
6237, 61eqsstrd 3631 . . . . . . 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 3028 . . . . . 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 7075 . . . . . . 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 396 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
67 onint 6980 . . . . . . 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 694 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
69 nfrab1 3117 . . . . . . . . 9 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7069nfint 4477 . . . . . . . 8 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
71 nfcv 2762 . . . . . . . 8 𝑦On
72 nfcv 2762 . . . . . . . . 9 𝑦𝐵
73 nfcv 2762 . . . . . . . . . 10 𝑦𝐴
74 nfcv 2762 . . . . . . . . . 10 𝑦 +𝑜
7573, 74, 70nfov 6661 . . . . . . . . 9 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7672, 75nfss 3588 . . . . . . . 8 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
77 oveq2 6643 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7877sseq2d 3625 . . . . . . . 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 3354 . . . . . . 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 480 . . . . . 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 3612 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵)
83 oveq2 6643 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
8483eqeq1d 2622 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵))
8584rspcev 3304 . . . 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 692 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
8786ex 450 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
88 nnaword1 7694 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
8988adantlr 750 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
90 sseq2 3619 . . . 4 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
9189, 90syl5ibcom 235 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
9291rexlimdva 3027 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
9387, 92impbid 202 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 196  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wrex 2910  {crab 2913  wss 3567  c0 3907   cint 4466  Ord word 5710  Oncon0 5711  suc csuc 5713  (class class class)co 6635  ωcom 7050   +𝑜 coa 7542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549
This theorem is referenced by:  nnaordex  7703  unfilem1  8209  hashdom  13151
  Copyright terms: Public domain W3C validator