Theorem o2p2e4 8478

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6331. For the usual proof using complex numbers, see 2p2e4 12287. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5226, 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 8420	. . . 4 ⊢ 2o ∈ On
2		df-1o 8407	. . . . 5 ⊢ 1o = suc ∅
3		peano1 7841	. . . . . 6 ⊢ ∅ ∈ ω
4		peano2 7842	. . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ suc ∅ ∈ ω
6	2, 5	eqeltri 2833	. . . 4 ⊢ 1o ∈ ω
7		onasuc 8465	. . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
8	1, 6, 7	mp2an 693	. . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o)
9		df-2o 8408	. . . 4 ⊢ 2o = suc 1o
10	9	oveq2i 7379	. . 3 ⊢ (2o +o 2o) = (2o +o suc 1o)
11		df-3o 8409	. . . . 5 ⊢ 3o = suc 2o
12		oa1suc 8468	. . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o)
13	1, 12	ax-mp 5	. . . . 5 ⊢ (2o +o 1o) = suc 2o
14	11, 13	eqtr4i 2763	. . . 4 ⊢ 3o = (2o +o 1o)
15		suceq 6393	. . . 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 2770	. 2 ⊢ (2o +o 2o) = suc 3o
18		df-4o 8410	. 2 ⊢ 4o = suc 3o
19	17, 18	eqtr4i 2763	1 ⊢ (2o +o 2o) = 4o

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∅c0 4287  Oncon0 6325  suc csuc 6327  (class class class)co 7368  ωcom 7818  1_oc1o 8400  2_oc2o 8401  3_oc3o 8402  4_oc4o 8403   +_o coa 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-3o 8409  df-4o 8410  df-oadd 8411

This theorem is referenced by: (None)