Theorem nnaordex2 8554

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 8553	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
2		nn0suc 7824	. . . . . . 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 4287	. . . . . . 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 7354	. . . . . . . 8 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
10	9	eqeq1d 2733	. . . . . . 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 3147	. . . . 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 3134	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
15		peano2 7820	. . . . . 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 7804	. . . . . . 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 7765	. . . . . 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 2820	. . . . . . 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 3572	. . . . 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 3134	. . 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 2111  ∃wrex 3056  ∅c0 4280  Ord word 6305  suc csuc 6308  (class class class)co 7346  ωcom 7796   +_o coa 8382

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389

This theorem is referenced by:  om2noseqlt  28229