Theorem o2p2e4 8451

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6316. For the usual proof using complex numbers, see 2p2e4 12218. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5237, 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 8390	. . . 4 ⊢ 2o ∈ On
2		df-1o 8376	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7812	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7814	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2834	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8438	. . . 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 8377	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7357	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8378	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8441	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2768	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6376	. . . 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 2775	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8379	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2768	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1541   ∈ wcel 2106  ∅c0 4277  Oncon0 6310  suc csuc 6312  (class class class)co 7346  ωcom 7789  1_oc1o 8369  2_oc2o 8370  3_oc3o 8371  4_oc4o 8372   +_o coa 8373

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379  ax-un 7659

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-tr 5218  df-id 5525  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-we 5584  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-pred 6246  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7790  df-2nd 7909  df-frecs 8176  df-wrecs 8207  df-recs 8281  df-rdg 8320  df-1o 8376  df-2o 8377  df-3o 8378  df-4o 8379  df-oadd 8380

This theorem is referenced by: (None)