Theorem o2p2e4OLD 8372

Description: Obsolete version of o2p2e4 8371 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 8311	. . . 4 ⊢ 2o ∈ On
2		1on 8309	. . . 4 ⊢ 1o ∈ On
3		oasuc 8354	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o suc 1o) = suc (2o +o 1o))
4	1, 2, 3	mp2an 689	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
5		df-2o 8298	. . . 4 ⊢ 2o = suc 1o
6	5	oveq2i 7286	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
7		df-3o 8299	. . . . 5 ⊢ 3o = suc 2o
8		oa1suc 8361	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
10	7, 9	eqtr4i 2769	. . . 4 ⊢ 3o = (2o +o 1o)
11		suceq 6331	. . . 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 2776	. 2 ⊢ (2o +o 2o) = suc 3o
14		df-4o 8300	. 2 ⊢ 4o = suc 3o
15	13, 14	eqtr4i 2769	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1539   ∈ wcel 2106  Oncon0 6266  suc csuc 6268  (class class class)co 7275  1_oc1o 8290  2_oc2o 8291  3_oc3o 8292  4_oc4o 8293   +_o coa 8294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-3o 8299  df-4o 8300  df-oadd 8301

This theorem is referenced by: (None)