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Mirrors > Home > MPE Home > Th. List > o2p2e4 | Structured version Visualization version Unicode version |
Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5729. For the usual proof using complex numbers, see 2p2e4 11144. (Contributed by NM, 18-Aug-2021.) |
Ref | Expression |
---|---|
o2p2e4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 7568 | . . . 4 | |
2 | 1on 7567 | . . . 4 | |
3 | oasuc 7604 | . . . 4 | |
4 | 1, 2, 3 | mp2an 708 | . . 3 |
5 | df-2o 7561 | . . . 4 | |
6 | 5 | oveq2i 6661 | . . 3 |
7 | df-3o 7562 | . . . . 5 | |
8 | oa1suc 7611 | . . . . . 6 | |
9 | 1, 8 | ax-mp 5 | . . . . 5 |
10 | 7, 9 | eqtr4i 2647 | . . . 4 |
11 | suceq 5790 | . . . 4 | |
12 | 10, 11 | ax-mp 5 | . . 3 |
13 | 4, 6, 12 | 3eqtr4i 2654 | . 2 |
14 | df-4o 7563 | . 2 | |
15 | 13, 14 | eqtr4i 2647 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 con0 5723 csuc 5725 (class class class)co 6650 c1o 7553 c2o 7554 c3o 7555 c4o 7556 coa 7557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-3o 7562 df-4o 7563 df-oadd 7564 |
This theorem is referenced by: (None) |
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