Theorem o2p2e4 8462

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6317. For the usual proof using complex numbers, see 2p2e4 12262. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5219, 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 8404	. . . 4 ⊢ 2o ∈ On
2		df-1o 8391	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7825	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7826	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2829	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8449	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 692	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8392	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7363	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8393	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8452	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2759	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6379	. . . 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 2766	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8394	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2759	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1541   ∈ wcel 2113  ∅c0 4282  Oncon0 6311  suc csuc 6313  (class class class)co 7352  ωcom 7802  1_oc1o 8384  2_oc2o 8385  3_oc3o 8386  4_oc4o 8387   +_o coa 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-3o 8393  df-4o 8394  df-oadd 8395

This theorem is referenced by: (None)