Theorem o2p2e4 8168

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6199. For the usual proof using complex numbers, see 2p2e4 11775. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5192, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)

Assertion

Ref	Expression
o2p2e4	⊢ (2o +o 2o) = 4o

Proof of Theorem o2p2e4

Step	Hyp	Ref	Expression
1		2on 8113	. . . 4 ⊢ 2o ∈ On
2		df-1o 8104	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7603	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7604	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2911	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8155	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 690	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8105	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7169	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8106	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8158	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2849	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6258	. . . 4 ⊢ (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o))
16	14, 15	ax-mp 5	. . 3 ⊢ suc 3o = suc (2o +o 1o)
17	8, 10, 16	3eqtr4i 2856	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8107	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2849	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∅c0 4293  Oncon0 6193  suc csuc 6195  (class class class)co 7158  ωcom 7582  1_oc1o 8097  2_oc2o 8098  3_oc3o 8099  4_oc4o 8100   +_o coa 8101

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-3o 8106  df-4o 8107  df-oadd 8108

This theorem is referenced by: (None)