Theorem o2p2e4 8476

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6329. For the usual proof using complex numbers, see 2p2e4 12311. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5212, 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 8418	. . . 4 ⊢ 2o ∈ On
2		df-1o 8405	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7840	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7841	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2832	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8463	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 693	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8406	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7378	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8407	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8466	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2762	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6391	. . . 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 2769	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8408	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2762	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∅c0 4273  Oncon0 6323  suc csuc 6325  (class class class)co 7367  ωcom 7817  1_oc1o 8398  2_oc2o 8399  3_oc3o 8400  4_oc4o 8401   +_o coa 8402

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-3o 8407  df-4o 8408  df-oadd 8409

This theorem is referenced by: (None)