Theorem o2p2e4 8535

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6373. For the usual proof using complex numbers, see 2p2e4 12393. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5243, 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 8476	. . . 4 ⊢ 2o ∈ On
2		df-1o 8462	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7894	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7895	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2862	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8522	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 705	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8463	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7434	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8464	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8525	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2792	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6436	. . . 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 2799	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8465	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2792	1 ⊢ (2o +o 2o) = 4o

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  ∅c0 4289  Oncon0 6367  suc csuc 6369  (class class class)co 7423  ωcom 7871  1_oc1o 8455  2_oc2o 8456  3_oc3o 8457  4_oc4o 8458   +_o coa 8459

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-3o 8464  df-4o 8465  df-oadd 8466

This theorem is used by: (None)