Theorem oneo 8518

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 6421	. . 3 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
2	1	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3		suceq 6393	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
4	3	eqeq1d 2739	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
5	4	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
6		ovex 7401	. . . . . . . 8 ⊢ (2o ·o 𝐴) ∈ V
7	6	sucid 6409	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
8		eleq2 2826	. . . . . . 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 233	. . . . . 6 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
10		2on 8420	. . . . . . . 8 ⊢ 2o ∈ On
11		omord 8505	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 2o ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
12	10, 11	mp3an3 1453	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
13		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) → 𝐴 ∈ 𝐵)
14	12, 13	biimtrrdi 254	. . . . . 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 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
17		omcl 8473	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o 𝐴) ∈ On)
18	10, 17	mpan 691	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o 𝐴) ∈ On)
19		oa1suc 8468	. . . . . . . . . . . 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 8417	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
22	21	sucid 6409	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
23		df-2o 8408	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
24	22, 23	eleqtrri 2836	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
25		1on 8419	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
26		oaord 8484	. . . . . . . . . . . . . 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 1468	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
28	24, 27	mpbii 233	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
29		omsuc 8463	. . . . . . . . . . . . 13 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
30	10, 29	mpan 691	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
31	28, 30	eleqtrrd 2840	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
32	20, 31	eqeltrrd 2838	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
33	32	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
34	16, 33	eqeltrrd 2838	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
35		onsuc 7765	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
36		omord 8505	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On ∧ 2o ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
37	10, 36	mp3an3 1453	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
38	35, 37	sylan2 594	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
39	38	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	39	adantr 480	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
42	41	simpld 494	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
43	42	ex 412	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
44	15, 43	jcad 512	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
45	44	3adant3 1133	. . 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 240	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
47	2, 46	mtod 198	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∅c0 4287  Oncon0 6325  suc csuc 6327  (class class class)co 7368  1_oc1o 8400  2_oc2o 8401   +_o coa 8404   ·_o comu 8405

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412

This theorem is referenced by: (None)