Theorem o2p2e4 8553

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 12481. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5231, 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 8494	. . . 4 ⊢ 2o ∈ On
2		df-1o 8480	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7901	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7902	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2856	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8540	. . . 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 8481	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7432	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8482	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8543	. . . . . 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 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 2793	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8483	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2786	1 ⊢ (2o +o 2o) = 4o

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∅c0 4278  Oncon0 6362  suc csuc 6364  (class class class)co 7421  ωcom 7878  1_oc1o 8473  2_oc2o 8474  3_oc3o 8475  4_oc4o 8476   +_o coa 8477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-3o 8482  df-4o 8483  df-oadd 8484

This theorem is used by: (None)