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Theorem nnawordex 8664
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 oveq2 7434 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
21sseq2d 4014 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
3 simplr 767 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
4 nnon 7882 . . . . . . . 8 (𝐵 ∈ ω → 𝐵 ∈ On)
53, 4syl 17 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ On)
6 simpll 765 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
7 nnaword2 8657 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵))
83, 6, 7syl2anc 582 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +o 𝐵))
92, 5, 8elrabd 3686 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
10 intss1 4970 . . . . . 6 (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
119, 10syl 17 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵)
12 ssrab2 4077 . . . . . . . 8 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On
139ne0d 4339 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅)
14 oninton 7804 . . . . . . . 8 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
1512, 13, 14sylancr 585 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
16 eloni 6384 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On → Ord {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
1715, 16syl 17 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
18 ordom 7886 . . . . . 6 Ord ω
19 ordtr2 6418 . . . . . 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))
2017, 18, 19sylancl 584 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω))
2111, 3, 20mp2and 697 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω)
22 nna0 8631 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
2322ad2antrr 724 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
24 simpr 483 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴𝐵)
2523, 24eqsstrd 4020 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o ∅) ⊆ 𝐵)
26 oveq2 7434 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = (𝐴 +o ∅))
2726sseq1d 4013 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → ((𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +o ∅) ⊆ 𝐵))
2825, 27syl5ibrcom 246 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵))
29 simprr 771 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
3029oveq2d 7442 . . . . . . . . 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))
316adantr 479 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
32 simprl 769 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
33 nnasuc 8633 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3431, 32, 33syl2anc 582 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3530, 34eqtrd 2768 . . . . . . . 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))
36 nnord 7884 . . . . . . . . . . 11 (𝐵 ∈ ω → Ord 𝐵)
373, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord 𝐵)
3837adantr 479 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord 𝐵)
39 nnon 7882 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
4039adantr 479 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
41 vex 3477 . . . . . . . . . . . . . 14 𝑥 ∈ V
4241sucid 6456 . . . . . . . . . . . . 13 𝑥 ∈ suc 𝑥
43 simpr 483 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
4442, 43eleqtrrid 2836 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
45 oveq2 7434 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
4645sseq2d 4014 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑥)))
4746onnminsb 7808 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +o 𝑥)))
4840, 44, 47sylc 65 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
4948adantl 480 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
50 nnacl 8638 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
5131, 32, 50syl2anc 582 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ ω)
52 nnord 7884 . . . . . . . . . . . . 13 ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
5351, 52syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord (𝐴 +o 𝑥))
54 ordtri1 6407 . . . . . . . . . . . 12 ((Ord 𝐵 ∧ Ord (𝐴 +o 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
5538, 53, 54syl2anc 582 . . . . . . . . . . 11 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
5655con2bid 353 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ((𝐴 +o 𝑥) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +o 𝑥)))
5749, 56mpbird 256 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ 𝐵)
58 ordsucss 7827 . . . . . . . . 9 (Ord 𝐵 → ((𝐴 +o 𝑥) ∈ 𝐵 → suc (𝐴 +o 𝑥) ⊆ 𝐵))
5938, 57, 58sylc 65 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → suc (𝐴 +o 𝑥) ⊆ 𝐵)
6035, 59eqsstrd 4020 . . . . . . 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)
6160rexlimdvaa 3153 . . . . . 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))
62 nn0suc 7907 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥))
6321, 62syl 17 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥))
6428, 61, 63mpjaod 858 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵)
65 onint 7799 . . . . . . 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)})
6612, 13, 65sylancr 585 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
67 nfrab1 3450 . . . . . . . . 9 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4963 . . . . . . . 8 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
69 nfcv 2899 . . . . . . . 8 𝑦On
70 nfcv 2899 . . . . . . . . 9 𝑦𝐵
71 nfcv 2899 . . . . . . . . . 10 𝑦𝐴
72 nfcv 2899 . . . . . . . . . 10 𝑦 +o
7371, 72, 68nfov 7456 . . . . . . . . 9 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7470, 73nfss 3974 . . . . . . . 8 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
75 oveq2 7434 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7675sseq2d 4014 . . . . . . . 8 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})))
7768, 69, 74, 76elrabf 3680 . . . . . . 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)})))
7877simprbi 495 . . . . . 6 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7966, 78syl 17 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
8064, 79eqssd 3999 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵)
81 oveq2 7434 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑥) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
8281eqeq1d 2730 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵))
8382rspcev 3611 . . . 4 (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω ∧ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
8421, 80, 83syl2anc 582 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
8584ex 411 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
86 nnaword1 8656 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
8786adantlr 713 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
88 sseq2 4008 . . . 4 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
8987, 88syl5ibcom 244 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵𝐴𝐵))
9089rexlimdva 3152 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵𝐴𝐵))
9185, 90impbid 211 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 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2937  wrex 3067  {crab 3430  wss 3949  c0 4326   cint 4953  Ord word 6373  Oncon0 6374  suc csuc 6376  (class class class)co 7426  ωcom 7876   +o coa 8490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-oadd 8497
This theorem is referenced by:  nnaordex  8665  eldifsucnn  8691  unfilem1  9341  ttrcltr  9747  hashdom  14378  precsexlem6  28130  precsexlem7  28131
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