Theorem nnaordex2 8579

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 8578	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵)))
2		nn0suc 7848	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
3	2	ad2antrl 729	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥))
4		simprrl 781	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∅ ∈ 𝑦)
5		n0i 4294	. . . . . . 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 879	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
8		simprrr 782	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ∈ ω ∧ (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))) → (𝐴 +o 𝑦) = 𝐵)
9		oveq2 7378	. . . . . . . 8 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
10	9	eqeq1d 2739	. . . . . . 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 3153	. . . . 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 3140	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
15		peano2 7844	. . . . . 6 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
16	15	ad2antrl 729	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → suc 𝑥 ∈ ω)
17		nnord 7828	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
18	17	ad2antrl 729	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → Ord 𝑥)
19		0elsuc 7789	. . . . . 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 773	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → (𝐴 +o suc 𝑥) = 𝐵)
22		eleq2 2826	. . . . . . 7 ⊢ (𝑦 = suc 𝑥 → (∅ ∈ 𝑦 ↔ ∅ ∈ suc 𝑥))
23	22, 10	anbi12d 633	. . . . . 6 ⊢ (𝑦 = suc 𝑥 → ((∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)))
24	23	rspcev 3578	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
25	16, 20, 21, 24	syl12anc 837	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑥 ∈ ω ∧ (𝐴 +o suc 𝑥) = 𝐵)) → ∃𝑦 ∈ ω (∅ ∈ 𝑦 ∧ (𝐴 +o 𝑦) = 𝐵))
26	25	rexlimdvaa 3140	. . 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 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ∅c0 4287  Ord word 6326  suc csuc 6329  (class class class)co 7370  ωcom 7820   +_o coa 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-oadd 8413

This theorem is referenced by:  om2noseqlt  28312