Theorem o2p2e4 8580

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6389. For the usual proof using complex numbers, see 2p2e4 12402. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5278, 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 8521	. . . 4 ⊢ 2o ∈ On
2		df-1o 8507	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7911	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7913	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2836	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8567	. . . 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 8508	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7443	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8509	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8570	. . . . . 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 6449	. . . 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 8510	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2767	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1539   ∈ wcel 2107  ∅c0 4332  Oncon0 6383  suc csuc 6385  (class class class)co 7432  ωcom 7888  1_oc1o 8500  2_oc2o 8501  3_oc3o 8502  4_oc4o 8503   +_o coa 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-3o 8509  df-4o 8510  df-oadd 8511

This theorem is referenced by: (None)