Theorem o2p2e4OLD 8167

Description: Obsolete version of o2p2e4 8166 as of 23-Mar-2024. (Contributed by NM, 18-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
o2p2e4OLD	⊢ (2o +o 2o) = 4o

Proof of Theorem o2p2e4OLD

Step	Hyp	Ref	Expression
1		2on 8111	. . . 4 ⊢ 2o ∈ On
2		1on 8109	. . . 4 ⊢ 1o ∈ On
3		oasuc 8149	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o suc 1o) = suc (2o +o 1o))
4	1, 2, 3	mp2an 690	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
5		df-2o 8103	. . . 4 ⊢ 2o = suc 1o
6	5	oveq2i 7167	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
7		df-3o 8104	. . . . 5 ⊢ 3o = suc 2o
8		oa1suc 8156	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
10	7, 9	eqtr4i 2847	. . . 4 ⊢ 3o = (2o +o 1o)
11		suceq 6256	. . . 4 ⊢ (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o))
12	10, 11	ax-mp 5	. . 3 ⊢ suc 3o = suc (2o +o 1o)
13	4, 6, 12	3eqtr4i 2854	. 2 ⊢ (2o +o 2o) = suc 3o
14		df-4o 8105	. 2 ⊢ 4o = suc 3o
15	13, 14	eqtr4i 2847	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1537   ∈ wcel 2114  Oncon0 6191  suc csuc 6193  (class class class)co 7156  1_oc1o 8095  2_oc2o 8096  3_oc3o 8097  4_oc4o 8098   +_o coa 8099

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-3o 8104  df-4o 8105  df-oadd 8106

This theorem is referenced by: (None)