Theorem oneo 8287

Description: If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)

Assertion

Ref	Expression
oneo	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Proof of Theorem oneo

Step	Hyp	Ref	Expression
1		onnbtwn 6282	. . 3 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
2	1	3ad2ant1 1135	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3		suceq 6256	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
4	3	eqeq1d 2738	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
5	4	3ad2ant3 1137	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
6		ovex 7224	. . . . . . . 8 ⊢ (2o ·o 𝐴) ∈ V
7	6	sucid 6270	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
8		eleq2 2819	. . . . . . 7 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
9	7, 8	mpbii 236	. . . . . 6 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
10		2on 8188	. . . . . . . 8 ⊢ 2o ∈ On
11		omord 8274	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2o ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
12	10, 11	mp3an3 1452	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
13		simpl 486	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) → 𝐴 ∈ 𝐵)
14	12, 13	syl6bir 257	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((2o ·o 𝐴) ∈ (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
15	9, 14	syl5 34	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
16		simpr 488	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
17		omcl 8241	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o 𝐴) ∈ On)
18	10, 17	mpan 690	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o 𝐴) ∈ On)
19		oa1suc 8236	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
20	18, 19	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
21		1oex 8193	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
22	21	sucid 6270	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
23		df-2o 8181	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
24	22, 23	eleqtrri 2830	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
25		1on 8187	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
26		oaord 8253	. . . . . . . . . . . . . 14 ⊢ ((1o ∈ On ∧ 2o ∈ On ∧ (2o ·o 𝐴) ∈ On) → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
27	25, 10, 18, 26	mp3an12i 1467	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
28	24, 27	mpbii 236	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
29		omsuc 8231	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
30	10, 29	mpan 690	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
31	28, 30	eleqtrrd 2834	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
32	20, 31	eqeltrrd 2832	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
33	32	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
34	16, 33	eqeltrrd 2832	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
35		suceloni 7570	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
36		omord 8274	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2o ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
37	10, 36	mp3an3 1452	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
38	35, 37	sylan2 596	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
39	38	ancoms 462	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	39	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
41	34, 40	mpbird 260	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
42	41	simpld 498	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
43	42	ex 416	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
44	15, 43	jcad 516	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
45	44	3adant3 1134	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
46	5, 45	sylbid 243	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
47	2, 46	mtod 201	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∅c0 4223  Oncon0 6191  suc csuc 6193  (class class class)co 7191  1_oc1o 8173  2_oc2o 8174   +_o coa 8177   ·_o comu 8178

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 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-2o 8181  df-oadd 8184  df-omul 8185

This theorem is referenced by: (None)