Theorem nnaordex2 8669

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 8668	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
2		nn0suc 7907	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
3	2	ad2antrl 726	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
4		simprrl 779	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∅ ∈ 𝑦)
5		n0i 4336	. . . . . . 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 876	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
8		simprrr 780	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝐴 +o 𝑦) = 𝐵)
9		oveq2 7432	. . . . . . . 8 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
10	9	eqeq1d 2728	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → ((𝐴 +o 𝑦) = 𝐵 ↔ (𝐴 +o suc 𝑥) = 𝐵))
11	8, 10	syl5ibcom 244	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = suc 𝑥 → (𝐴 +o suc 𝑥) = 𝐵))
12	11	reximdv 3160	. . . . 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 3146	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
15		peano2 7902	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
16	15	ad2antrl 726	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → suc 𝑥 ∈ ω)
17		nnord 7884	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
18	17	ad2antrl 726	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → Ord 𝑥)
19		0elsuc 7844	. . . . . 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 771	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → (𝐴 +o suc 𝑥) = 𝐵)
22		eleq2 2815	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → (∅ ∈ 𝑦 ↔ ∅ ∈ suc 𝑥))
23	22, 10	anbi12d 630	. . . . . 6 ⊢ (𝑦 = suc 𝑥 → ((∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)))
24	23	rspcev 3608	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
25	16, 20, 21, 24	syl12anc 835	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
26	25	rexlimdvaa 3146	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵 → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
27	14, 26	impbid 211	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
28	1, 27	bitrd 278	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  ∅c0 4325  Ord word 6375  suc csuc 6378  (class class class)co 7424  ωcom 7876   +_o coa 8493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  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 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  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 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-oadd 8500

This theorem is referenced by:  om2noseqlt  28273