Theorem o2p2e4 8514

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6356. For the usual proof using complex numbers, see 2p2e4 12366. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5232, 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 8455	. . . 4 ⊢ 2o ∈ On
2		df-1o 8441	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7873	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7874	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2861	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8501	. . . 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 8442	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7411	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8443	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8504	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2791	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6418	. . . 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 2798	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8444	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2791	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1563   ∈ wcel 2145  ∅c0 4288  Oncon0 6350  suc csuc 6352  (class class class)co 7400  ωcom 7850  1_oc1o 8434  2_oc2o 8435  3_oc3o 8436  4_oc4o 8437   +_o coa 8438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-3o 8443  df-4o 8444  df-oadd 8445

This theorem is referenced by: (None)