Theorem o2p2e4 8597

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6401. For the usual proof using complex numbers, see 2p2e4 12428. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5303, 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 8536	. . . 4 ⊢ 2o ∈ On
2		df-1o 8522	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7927	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7929	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2840	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8584	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 691	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8523	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7459	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8524	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8587	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2771	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6461	. . . 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 2778	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8525	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2771	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1537   ∈ wcel 2108  ∅c0 4352  Oncon0 6395  suc csuc 6397  (class class class)co 7448  ωcom 7903  1_oc1o 8515  2_oc2o 8516  3_oc3o 8517  4_oc4o 8518   +_o coa 8519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-3o 8524  df-4o 8525  df-oadd 8526

This theorem is referenced by: (None)