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Theorem o2p2e4 8155
Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6190. For the usual proof using complex numbers, see 2p2e4 11760. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5181, 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 8100 . . . 4 2o ∈ On
2 df-1o 8091 . . . . 5 1o = suc ∅
3 peano1 7590 . . . . . 6 ∅ ∈ ω
4 peano2 7591 . . . . . 6 (∅ ∈ ω → suc ∅ ∈ ω)
53, 4ax-mp 5 . . . . 5 suc ∅ ∈ ω
62, 5eqeltri 2906 . . . 4 1o ∈ ω
7 onasuc 8142 . . . 4 ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o))
81, 6, 7mp2an 688 . . 3 (2o +o suc 1o) = suc (2o +o 1o)
9 df-2o 8092 . . . 4 2o = suc 1o
109oveq2i 7156 . . 3 (2o +o 2o) = (2o +o suc 1o)
11 df-3o 8093 . . . . 5 3o = suc 2o
12 oa1suc 8145 . . . . . 6 (2o ∈ On → (2o +o 1o) = suc 2o)
131, 12ax-mp 5 . . . . 5 (2o +o 1o) = suc 2o
1411, 13eqtr4i 2844 . . . 4 3o = (2o +o 1o)
15 suceq 6249 . . . 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 2851 . 2 (2o +o 2o) = suc 3o
18 df-4o 8094 . 2 4o = suc 3o
1917, 18eqtr4i 2844 1 (2o +o 2o) = 4o
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  c0 4288  Oncon0 6184  suc csuc 6186  (class class class)co 7145  ωcom 7569  1oc1o 8084  2oc2o 8085  3oc3o 8086  4oc4o 8087   +o coa 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-3o 8093  df-4o 8094  df-oadd 8095
This theorem is referenced by: (None)
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