Theorem o2p2e4 8587

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6394. For the usual proof using complex numbers, see 2p2e4 12517. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5238, 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 8528	. . . 4 ⊢ 2o ∈ On
2		df-1o 8514	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7934	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7935	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2860	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8574	. . . 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 8515	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7462	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8516	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8577	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2789	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6457	. . . 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 2796	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8517	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2789	1 ⊢ (2o +o 2o) = 4o

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∅c0 4281  Oncon0 6388  suc csuc 6390  (class class class)co 7451  ωcom 7911  1_oc1o 8507  2_oc2o 8508  3_oc3o 8509  4_oc4o 8510   +_o coa 8511

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518

This theorem is used by: (None)