Theorem oneo 8617

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 6477	. . 3 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
2	1	3ad2ant1 1151	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3		suceq 6449	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
4	3	eqeq1d 2765	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
5	4	3ad2ant3 1153	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
6		ovex 7476	. . . . . . . 8 ⊢ (2o ·o 𝐴) ∈ V
7	6	sucid 6465	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
8		eleq2 2853	. . . . . . 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 8518	. . . . . . . 8 ⊢ 2o ∈ On
11		omord 8604	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2o ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
12	10, 11	mp3an3 1480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
13		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) → 𝐴 ∈ 𝐵)
14	12, 13	biimtrrdi 257	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((2o ·o 𝐴) ∈ (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
15	9, 14	syl5 35	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
16		simpr 490	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
17		omcl 8572	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o 𝐴) ∈ On)
18	10, 17	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o 𝐴) ∈ On)
19		oa1suc 8567	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
20	18, 19	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
21		1oex 8514	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
22	21	sucid 6465	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
23		df-2o 8505	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
24	22, 23	eleqtrri 2863	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
25		1on 8517	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
26		oaord 8583	. . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . 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 8562	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
30	10, 29	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
31	28, 30	eleqtrrd 2867	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
32	20, 31	eqeltrrd 2865	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
33	32	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
34	16, 33	eqeltrrd 2865	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
35		onsuc 7848	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
36		omord 8604	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2o ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
37	10, 36	mp3an3 1480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
38	35, 37	sylan2 605	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
39	38	ancoms 464	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	39	adantr 486	. . . . . . . 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 500	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
43	42	ex 418	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
44	15, 43	jcad 522	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
45	44	3adant3 1150	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∅c0 4281  Oncon0 6380  suc csuc 6382  (class class class)co 7443  1_oc1o 8497  2_oc2o 8498   +_o coa 8501   ·_o comu 8502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509

This theorem is used by: (None)