Theorem o2p2e4 8542

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6367. For the usual proof using complex numbers, see 2p2e4 12470. (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 8483	. . . 4 ⊢ 2o ∈ On
2		df-1o 8469	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7898	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7899	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2857	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8529	. . . 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 8470	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7429	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8471	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8532	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2787	. . . 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 2794	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8472	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2787	1 ⊢ (2o +o 2o) = 4o

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∅c0 4279  Oncon0 6361  suc csuc 6363  (class class class)co 7418  ωcom 7875  1_oc1o 8462  2_oc2o 8463  3_oc3o 8464  4_oc4o 8465   +_o coa 8466

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-3o 8471  df-4o 8472  df-oadd 8473

This theorem is used by: (None)