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Theorem nnawordex 7870
 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 744 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
2 nnon 7217 . . . . . . . 8 (𝐵 ∈ ω → 𝐵 ∈ On)
31, 2syl 17 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ On)
4 simpll 742 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
5 nnaword2 7863 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
61, 4, 5syl2anc 565 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
7 oveq2 6800 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
87sseq2d 3780 . . . . . . . 8 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
98elrab 3513 . . . . . . 7 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
103, 6, 9sylanbrc 564 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
11 intss1 4624 . . . . . 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 3834 . . . . . . . 8 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
14 ne0i 4067 . . . . . . . . 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 7146 . . . . . . . 8 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
1713, 15, 16sylancr 567 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
18 eloni 5876 . . . . . . 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 7220 . . . . . 6 Ord ω
21 ordtr2 5911 . . . . . 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 566 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
2312, 1, 22mp2and 671 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω)
24 nna0 7837 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
2524ad2antrr 697 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
26 simpr 471 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴𝐵)
2725, 26eqsstrd 3786 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ⊆ 𝐵)
28 oveq2 6800 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 ∅))
2928sseq1d 3779 . . . . . . 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 748 . . . . . . . . . 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 6808 . . . . . . . . 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 466 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
34 simprl 746 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
35 nnasuc 7839 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3633, 34, 35syl2anc 565 . . . . . . . . 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 2804 . . . . . . . 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 7219 . . . . . . . . . . 11 (𝐵 ∈ ω → Ord 𝐵)
391, 38syl 17 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord 𝐵)
4039adantr 466 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord 𝐵)
41 nnon 7217 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
4241adantr 466 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
43 vex 3352 . . . . . . . . . . . . . 14 𝑥 ∈ V
4443sucid 5947 . . . . . . . . . . . . 13 𝑥 ∈ suc 𝑥
45 simpr 471 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
4644, 45syl5eleqr 2856 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
47 oveq2 6800 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑥))
4847sseq2d 3780 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
4948onnminsb 7150 . . . . . . . . . . . 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 467 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
52 nnacl 7844 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
5333, 34, 52syl2anc 565 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ ω)
54 nnord 7219 . . . . . . . . . . . . 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 5899 . . . . . . . . . . . 12 ((Ord 𝐵 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
5740, 55, 56syl2anc 565 . . . . . . . . . . 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 343 . . . . . . . . . 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 7164 . . . . . . . . 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 3786 . . . . . . 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 3179 . . . . . 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 7236 . . . . . . 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 840 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
67 onint 7141 . . . . . . 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 567 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
69 nfrab1 3270 . . . . . . . . 9 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
7069nfint 4619 . . . . . . . 8 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
71 nfcv 2912 . . . . . . . 8 𝑦On
72 nfcv 2912 . . . . . . . . 9 𝑦𝐵
73 nfcv 2912 . . . . . . . . . 10 𝑦𝐴
74 nfcv 2912 . . . . . . . . . 10 𝑦 +𝑜
7573, 74, 70nfov 6820 . . . . . . . . 9 𝑦(𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
7672, 75nfss 3743 . . . . . . . 8 𝑦 𝐵 ⊆ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
77 oveq2 6800 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
7877sseq2d 3780 . . . . . . . 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 3509 . . . . . . 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 3767 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵)
83 oveq2 6800 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
8483eqeq1d 2772 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵))
8584rspcev 3458 . . . 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 565 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
8786ex 397 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
88 nnaword1 7862 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
8988adantlr 686 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
90 sseq2 3774 . . . 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 3178 . 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   ∧ wa 382   ∨ wo 826   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  ∃wrex 3061  {crab 3064   ⊆ wss 3721  ∅c0 4061  ∩ cint 4609  Ord word 5865  Oncon0 5866  suc csuc 5868  (class class class)co 6792  ωcom 7211   +𝑜 coa 7709 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-oadd 7716 This theorem is referenced by:  nnaordex  7871  unfilem1  8379  hashdom  13369
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