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Theorem nnawordex 8455
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 7275 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
21sseq2d 3952 . . . . . . 7 (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝐵)))
3 simplr 766 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
4 nnon 7708 . . . . . . . 8 (𝐵 ∈ ω → 𝐵 ∈ On)
53, 4syl 17 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ On)
6 simpll 764 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
7 nnaword2 8448 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +o 𝐵))
83, 6, 7syl2anc 584 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +o 𝐵))
92, 5, 8elrabd 3625 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
10 intss1 4894 . . . . . 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 4012 . . . . . . . 8 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On
139ne0d 4269 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅)
14 oninton 7635 . . . . . . . 8 (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
1512, 13, 14sylancr 587 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ On)
16 eloni 6269 . . . . . . 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 7712 . . . . . 6 Ord ω
19 ordtr2 6303 . . . . . 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 586 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⊆ 𝐵𝐵 ∈ ω) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω))
2111, 3, 20mp2and 696 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ ω)
22 nna0 8422 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
2322ad2antrr 723 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
24 simpr 485 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴𝐵)
2523, 24eqsstrd 3958 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o ∅) ⊆ 𝐵)
26 oveq2 7275 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = ∅ → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = (𝐴 +o ∅))
2726sseq1d 3951 . . . . . . 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 770 . . . . . . . . . 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 7283 . . . . . . . . 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 481 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
32 simprl 768 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
33 nnasuc 8424 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3431, 32, 33syl2anc 584 . . . . . . . . 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 2778 . . . . . . . 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 7710 . . . . . . . . . . 11 (𝐵 ∈ ω → Ord 𝐵)
373, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → Ord 𝐵)
3837adantr 481 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → Ord 𝐵)
39 nnon 7708 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
4039adantr 481 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
41 vex 3433 . . . . . . . . . . . . . 14 𝑥 ∈ V
4241sucid 6338 . . . . . . . . . . . . 13 𝑥 ∈ suc 𝑥
43 simpr 485 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)
4442, 43eleqtrrid 2846 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥) → 𝑥 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
45 oveq2 7275 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
4645sseq2d 3952 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +o 𝑦) ↔ 𝐵 ⊆ (𝐴 +o 𝑥)))
4746onnminsb 7639 . . . . . . . . . . . 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 482 . . . . . . . . . 10 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +o 𝑥))
50 nnacl 8429 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
5131, 32, 50syl2anc 584 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ (𝑥 ∈ ω ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} = suc 𝑥)) → (𝐴 +o 𝑥) ∈ ω)
52 nnord 7710 . . . . . . . . . . . . 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 6292 . . . . . . . . . . . 12 ((Ord 𝐵 ∧ Ord (𝐴 +o 𝑥)) → (𝐵 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐵))
5538, 53, 54syl2anc 584 . . . . . . . . . . 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 355 . . . . . . . . . 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 7655 . . . . . . . . 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 3958 . . . . . . 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 3212 . . . . . 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 7732 . . . . . . 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 857 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) ⊆ 𝐵)
65 onint 7630 . . . . . . 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 587 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
67 nfrab1 3315 . . . . . . . . 9 𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
6867nfint 4889 . . . . . . . 8 𝑦 {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}
69 nfcv 2907 . . . . . . . 8 𝑦On
70 nfcv 2907 . . . . . . . . 9 𝑦𝐵
71 nfcv 2907 . . . . . . . . . 10 𝑦𝐴
72 nfcv 2907 . . . . . . . . . 10 𝑦 +o
7371, 72, 68nfov 7297 . . . . . . . . 9 𝑦(𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
7470, 73nfss 3912 . . . . . . . 8 𝑦 𝐵 ⊆ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)})
75 oveq2 7275 . . . . . . . . 9 (𝑦 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑦) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
7675sseq2d 3952 . . . . . . . 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 3619 . . . . . . 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 497 . . . . . 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 3937 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵)
81 oveq2 7275 . . . . . 6 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → (𝐴 +o 𝑥) = (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}))
8281eqeq1d 2740 . . . . 5 (𝑥 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)}) = 𝐵))
8382rspcev 3559 . . . 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 584 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
8584ex 413 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
86 nnaword1 8447 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
8786adantlr 712 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
88 sseq2 3946 . . . 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 3211 . 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 396  wo 844   = wceq 1539  wcel 2106  wne 2943  wrex 3065  {crab 3068  wss 3886  c0 4256   cint 4879  Ord word 6258  Oncon0 6259  suc csuc 6261  (class class class)co 7267  ωcom 7702   +o coa 8281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-om 7703  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-oadd 8288
This theorem is referenced by:  nnaordex  8456  eldifsucnn  8481  unfilem1  9065  ttrcltr  9461  hashdom  14104
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