Theorem o2p2e4 8547

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6370. For the usual proof using complex numbers, see 2p2e4 12354. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5285, 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 8486	. . . 4 ⊢ 2o ∈ On
2		df-1o 8472	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7883	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7885	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2828	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8534	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 689	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8473	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7423	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8474	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8537	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2762	. . . 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 2769	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8475	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2762	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1540   ∈ wcel 2105  ∅c0 4322  Oncon0 6364  suc csuc 6366  (class class class)co 7412  ωcom 7859  1_oc1o 8465  2_oc2o 8466  3_oc3o 8467  4_oc4o 8468   +_o coa 8469

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-3o 8474  df-4o 8475  df-oadd 8476

This theorem is referenced by: (None)