Theorem o2p2e4 8558

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6363. For the usual proof using complex numbers, see 2p2e4 12380. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5254, 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 8499	. . . 4 ⊢ 2o ∈ On
2		df-1o 8485	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7889	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7891	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2831	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8545	. . . 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 8486	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7421	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8487	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8548	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2762	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6424	. . . 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 2769	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8488	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2762	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∅c0 4313  Oncon0 6357  suc csuc 6359  (class class class)co 7410  ωcom 7866  1_oc1o 8478  2_oc2o 8479  3_oc3o 8480  4_oc4o 8481   +_o coa 8482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-3o 8487  df-4o 8488  df-oadd 8489

This theorem is referenced by: (None)