Theorem o2p2e4 8402

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6287. For the usual proof using complex numbers, see 2p2e4 12154. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5218, 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 8342	. . . 4 ⊢ 2o ∈ On
2		df-1o 8328	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7767	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7769	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2833	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8389	. . . 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 8329	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7318	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8330	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8392	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2767	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6346	. . . 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 2774	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8331	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2767	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1539   ∈ wcel 2104  ∅c0 4262  Oncon0 6281  suc csuc 6283  (class class class)co 7307  ωcom 7744  1_oc1o 8321  2_oc2o 8322  3_oc3o 8323  4_oc4o 8324   +_o coa 8325

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-2o 8329  df-3o 8330  df-4o 8331  df-oadd 8332

This theorem is referenced by: (None)