Theorem o2p2e4 8508

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6341. For the usual proof using complex numbers, see 2p2e4 12323. (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 8450	. . . 4 ⊢ 2o ∈ On
2		df-1o 8437	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7868	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7869	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2825	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8495	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 692	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8438	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7401	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8439	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8498	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2756	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6403	. . . 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 2763	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8440	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2756	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∅c0 4299  Oncon0 6335  suc csuc 6337  (class class class)co 7390  ωcom 7845  1_oc1o 8430  2_oc2o 8431  3_oc3o 8432  4_oc4o 8433   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-3o 8439  df-4o 8440  df-oadd 8441

This theorem is referenced by: (None)