Theorem o2p2e4 8527

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6368. For the usual proof using complex numbers, see 2p2e4 12376. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5239, 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 8468	. . . 4 ⊢ 2o ∈ On
2		df-1o 8454	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7886	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7887	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2859	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8514	. . . 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 8455	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7423	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8456	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8517	. . . . . 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 6431	. . . 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 8457	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2789	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1570   ∈ wcel 2143  ∅c0 4287  Oncon0 6362  suc csuc 6364  (class class class)co 7412  ωcom 7863  1_oc1o 8447  2_oc2o 8448  3_oc3o 8449  4_oc4o 8450   +_o coa 8451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-3o 8456  df-4o 8457  df-oadd 8458

This theorem is referenced by: (None)