Theorem o2p2e4 8528

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6363. For the usual proof using complex numbers, see 2p2e4 12399. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5232, 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 8469	. . . 4 ⊢ 2o ∈ On
2		df-1o 8455	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7885	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7886	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2856	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8515	. . . 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 8456	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7424	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8457	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8518	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2786	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6426	. . . 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 2793	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8458	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2786	1 ⊢ (2o +o 2o) = 4o

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∅c0 4279  Oncon0 6357  suc csuc 6359  (class class class)co 7413  ωcom 7862  1_oc1o 8448  2_oc2o 8449  3_oc3o 8450  4_oc4o 8451   +_o coa 8452

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-3o 8457  df-4o 8458  df-oadd 8459

This theorem is used by: (None)