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Theorem o2p2e4 7566
 Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5688. For the usual proof using complex numbers, see 2p2e4 11088. (Contributed by NM, 18-Aug-2021.)
Assertion
Ref Expression
o2p2e4 (2𝑜 +𝑜 2𝑜) = 4𝑜

Proof of Theorem o2p2e4
StepHypRef Expression
1 2on 7513 . . . 4 2𝑜 ∈ On
2 1on 7512 . . . 4 1𝑜 ∈ On
3 oasuc 7549 . . . 4 ((2𝑜 ∈ On ∧ 1𝑜 ∈ On) → (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜))
41, 2, 3mp2an 707 . . 3 (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜)
5 df-2o 7506 . . . 4 2𝑜 = suc 1𝑜
65oveq2i 6615 . . 3 (2𝑜 +𝑜 2𝑜) = (2𝑜 +𝑜 suc 1𝑜)
7 df-3o 7507 . . . . 5 3𝑜 = suc 2𝑜
8 oa1suc 7556 . . . . . 6 (2𝑜 ∈ On → (2𝑜 +𝑜 1𝑜) = suc 2𝑜)
91, 8ax-mp 5 . . . . 5 (2𝑜 +𝑜 1𝑜) = suc 2𝑜
107, 9eqtr4i 2646 . . . 4 3𝑜 = (2𝑜 +𝑜 1𝑜)
11 suceq 5749 . . . 4 (3𝑜 = (2𝑜 +𝑜 1𝑜) → suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜))
1210, 11ax-mp 5 . . 3 suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜)
134, 6, 123eqtr4i 2653 . 2 (2𝑜 +𝑜 2𝑜) = suc 3𝑜
14 df-4o 7508 . 2 4𝑜 = suc 3𝑜
1513, 14eqtr4i 2646 1 (2𝑜 +𝑜 2𝑜) = 4𝑜
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  Oncon0 5682  suc csuc 5684  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499  3𝑜c3o 7500  4𝑜c4o 7501   +𝑜 coa 7502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-3o 7507  df-4o 7508  df-oadd 7509 This theorem is referenced by: (None)
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