Theorem o2p2e4 8571

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6382. For the usual proof using complex numbers, see 2p2e4 12399. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5290, 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 8510	. . . 4 ⊢ 2o ∈ On
2		df-1o 8496	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7900	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7902	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2822	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8558	. . . 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 8497	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7435	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8498	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8561	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2757	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6442	. . . 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 2764	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8499	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2757	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1534   ∈ wcel 2099  ∅c0 4325  Oncon0 6376  suc csuc 6378  (class class class)co 7424  ωcom 7876  1_oc1o 8489  2_oc2o 8490  3_oc3o 8491  4_oc4o 8492   +_o coa 8493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-3o 8498  df-4o 8499  df-oadd 8500

This theorem is referenced by: (None)