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Theorem o2p2e4 8153
Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6175. For the usual proof using complex numbers, see 2p2e4 11760. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5166, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
Assertion
Ref Expression
o2p2e4 (2o +o 2o) = 4o

Proof of Theorem o2p2e4
StepHypRef Expression
1 2on 8098 . . . 4 2o ∈ On
2 df-1o 8089 . . . . 5 1o = suc ∅
3 peano1 7586 . . . . . 6 ∅ ∈ ω
4 peano2 7587 . . . . . 6 (∅ ∈ ω → suc ∅ ∈ ω)
53, 4ax-mp 5 . . . . 5 suc ∅ ∈ ω
62, 5eqeltri 2910 . . . 4 1o ∈ ω
7 onasuc 8140 . . . 4 ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
81, 6, 7mp2an 691 . . 3 (2o +o suc 1o) = suc (2o +o 1o)
9 df-2o 8090 . . . 4 2o = suc 1o
109oveq2i 7151 . . 3 (2o +o 2o) = (2o +o suc 1o)
11 df-3o 8091 . . . . 5 3o = suc 2o
12 oa1suc 8143 . . . . . 6 (2o ∈ On → (2o +o 1o) = suc 2o)
131, 12ax-mp 5 . . . . 5 (2o +o 1o) = suc 2o
1411, 13eqtr4i 2848 . . . 4 3o = (2o +o 1o)
15 suceq 6234 . . . 4 (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o))
1614, 15ax-mp 5 . . 3 suc 3o = suc (2o +o 1o)
178, 10, 163eqtr4i 2855 . 2 (2o +o 2o) = suc 3o
18 df-4o 8092 . 2 4o = suc 3o
1917, 18eqtr4i 2848 1 (2o +o 2o) = 4o
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  c0 4265  Oncon0 6169  suc csuc 6171  (class class class)co 7140  ωcom 7565  1oc1o 8082  2oc2o 8083  3oc3o 8084  4oc4o 8085   +o coa 8086
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-3o 8091  df-4o 8092  df-oadd 8093
This theorem is referenced by: (None)
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