Theorem nnaordex2 8569

Description: Equivalence for ordering. (Contributed by Scott Fenton, 18-Apr-2025.)

Assertion

Ref	Expression
nnaordex2	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnaordex2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnaordex 8568	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
2		nn0suc 7838	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
3	2	ad2antrl 735	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
4		simprrl 787	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∅ ∈ 𝑦)
5		n0i 4271	. . . . . . 7 ⊢ (∅ ∈ 𝑦 → ¬ 𝑦 = ∅)
6	4, 5	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ¬ 𝑦 = ∅)
7	3, 6	orcnd 885	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
8		simprrr 788	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝐴 +o 𝑦) = 𝐵)
9		oveq2 7368	. . . . . . . 8 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
10	9	eqeq1d 2743	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → ((𝐴 +o 𝑦) = 𝐵 ↔ (𝐴 +o suc 𝑥) = 𝐵))
11	8, 10	syl5ibcom 247	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = suc 𝑥 → (𝐴 +o suc 𝑥) = 𝐵))
12	11	reximdv 3156	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
13	7, 12	mpd 15	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵)
14	13	rexlimdvaa 3143	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
15		peano2 7834	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
16	15	ad2antrl 735	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → suc 𝑥 ∈ ω)
17		nnord 7818	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
18	17	ad2antrl 735	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → Ord 𝑥)
19		0elsuc 7779	. . . . . 6 ⊢ (Ord 𝑥 → ∅ ∈ suc 𝑥)
20	18, 19	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∅ ∈ suc 𝑥)
21		simprr 779	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → (𝐴 +o suc 𝑥) = 𝐵)
22		eleq2 2830	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → (∅ ∈ 𝑦 ↔ ∅ ∈ suc 𝑥))
23	22, 10	anbi12d 639	. . . . . 6 ⊢ (𝑦 = suc 𝑥 → ((∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)))
24	23	rspcev 3562	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
25	16, 20, 21, 24	syl12anc 843	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
26	25	rexlimdvaa 3143	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵 → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
27	14, 26	impbid 214	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
28	1, 27	bitrd 281	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  ∅c0 4264  Ord word 6313  suc csuc 6316  (class class class)co 7360  ωcom 7810   +_o coa 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-oadd 8403

This theorem is referenced by:  om2noseqlt  28313  mh-inf3f1  36784