Theorem o2p2e4 8149

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6165. For the usual proof using complex numbers, see 2p2e4 11760. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5154, 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 8094	. . . 4 ⊢ 2o ∈ On
2		df-1o 8085	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7581	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7582	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2886	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8136	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 691	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8086	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7146	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8087	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8139	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2824	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6224	. . . 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 2831	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8088	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2824	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1538   ∈ wcel 2111  ∅c0 4243  Oncon0 6159  suc csuc 6161  (class class class)co 7135  ωcom 7560  1_oc1o 8078  2_oc2o 8079  3_oc3o 8080  4_oc4o 8081   +_o coa 8082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-3o 8087  df-4o 8088  df-oadd 8089

This theorem is referenced by: (None)