Theorem o2p2e4 7889

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5970. For the usual proof using complex numbers, see 2p2e4 11494. (Contributed by NM, 18-Aug-2021.)

Assertion

Ref	Expression
o2p2e4	⊢ (2o +o 2o) = 4o

Proof of Theorem o2p2e4

Step	Hyp	Ref	Expression
1		2on 7836	. . . 4 ⊢ 2o ∈ On
2		1on 7834	. . . 4 ⊢ 1o ∈ On
3		oasuc 7872	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o suc 1o) = suc (2o +o 1o))
4	1, 2, 3	mp2an 685	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
5		df-2o 7828	. . . 4 ⊢ 2o = suc 1o
6	5	oveq2i 6917	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
7		df-3o 7829	. . . . 5 ⊢ 3o = suc 2o
8		oa1suc 7879	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
10	7, 9	eqtr4i 2853	. . . 4 ⊢ 3o = (2o +o 1o)
11		suceq 6029	. . . 4 ⊢ (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o))
12	10, 11	ax-mp 5	. . 3 ⊢ suc 3o = suc (2o +o 1o)
13	4, 6, 12	3eqtr4i 2860	. 2 ⊢ (2o +o 2o) = suc 3o
14		df-4o 7830	. 2 ⊢ 4o = suc 3o
15	13, 14	eqtr4i 2853	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1658   ∈ wcel 2166  Oncon0 5964  suc csuc 5966  (class class class)co 6906  1_oc1o 7820  2_oc2o 7821  3_oc3o 7822  4_oc4o 7823   +_o coa 7824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-2o 7828  df-3o 7829  df-4o 7830  df-oadd 7831

This theorem is referenced by: (None)