Theorem o2p2e4 8525

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6367. For the usual proof using complex numbers, see 2p2e4 12374. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5242, 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 8466	. . . 4 ⊢ 2o ∈ On
2		df-1o 8452	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7884	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7885	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2865	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8512	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 704	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8453	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7422	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8454	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8515	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2795	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6430	. . . 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 2802	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8455	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2795	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1567   ∈ wcel 2149  ∅c0 4294  Oncon0 6361  suc csuc 6363  (class class class)co 7411  ωcom 7861  1_oc1o 8445  2_oc2o 8446  3_oc3o 8447  4_oc4o 8448   +_o coa 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-3o 8454  df-4o 8455  df-oadd 8456

This theorem is referenced by: (None)