Theorem nnaordex2 8565

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 8564	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
2		nn0suc 7834	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
3	2	ad2antrl 728	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
4		simprrl 780	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∅ ∈ 𝑦)
5		n0i 4290	. . . . . . 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 878	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
8		simprrr 781	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝐴 +o 𝑦) = 𝐵)
9		oveq2 7364	. . . . . . . 8 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
10	9	eqeq1d 2736	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → ((𝐴 +o 𝑦) = 𝐵 ↔ (𝐴 +o suc 𝑥) = 𝐵))
11	8, 10	syl5ibcom 245	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = suc 𝑥 → (𝐴 +o suc 𝑥) = 𝐵))
12	11	reximdv 3149	. . . . 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 3136	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
15		peano2 7830	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
16	15	ad2antrl 728	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → suc 𝑥 ∈ ω)
17		nnord 7814	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
18	17	ad2antrl 728	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → Ord 𝑥)
19		0elsuc 7775	. . . . . 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 772	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → (𝐴 +o suc 𝑥) = 𝐵)
22		eleq2 2823	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → (∅ ∈ 𝑦 ↔ ∅ ∈ suc 𝑥))
23	22, 10	anbi12d 632	. . . . . 6 ⊢ (𝑦 = suc 𝑥 → ((∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)))
24	23	rspcev 3574	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
25	16, 20, 21, 24	syl12anc 836	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
26	25	rexlimdvaa 3136	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵 → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
27	14, 26	impbid 212	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
28	1, 27	bitrd 279	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  ∅c0 4283  Ord word 6314  suc csuc 6317  (class class class)co 7356  ωcom 7806   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399

This theorem is referenced by:  om2noseqlt  28260