Theorem o2p2e4 8347

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6269. For the usual proof using complex numbers, see 2p2e4 12091. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5213, 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 8289	. . . 4 ⊢ 2o ∈ On
2		df-1o 8281	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7723	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7724	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2836	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8334	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 688	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8282	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7279	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8283	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8337	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2770	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6328	. . . 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 2777	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8284	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2770	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1541   ∈ wcel 2109  ∅c0 4261  Oncon0 6263  suc csuc 6265  (class class class)co 7268  ωcom 7700  1_oc1o 8274  2_oc2o 8275  3_oc3o 8276  4_oc4o 8277   +_o coa 8278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-2o 8282  df-3o 8283  df-4o 8284  df-oadd 8285

This theorem is referenced by: (None)