Theorem o2p2e4 8456

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6312. For the usual proof using complex numbers, see 2p2e4 12255. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5217, 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 8398	. . . 4 ⊢ 2o ∈ On
2		df-1o 8385	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7819	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7820	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2827	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8443	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 692	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8386	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7357	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8387	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8446	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2757	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6374	. . . 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 2764	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8388	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2757	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1541   ∈ wcel 2111  ∅c0 4283  Oncon0 6306  suc csuc 6308  (class class class)co 7346  ωcom 7796  1_oc1o 8378  2_oc2o 8379  3_oc3o 8380  4_oc4o 8381   +_o coa 8382

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-3o 8387  df-4o 8388  df-oadd 8389

This theorem is referenced by: (None)