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Mirrors > Home > MPE Home > Th. List > o1p1e2 | Structured version Visualization version GIF version |
Description: 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
o1p1e2 | ⊢ (1o +o 1o) = 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8460 | . . 3 ⊢ 1o ∈ On | |
2 | oa1suc 8513 | . . 3 ⊢ (1o ∈ On → (1o +o 1o) = suc 1o) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (1o +o 1o) = suc 1o |
4 | df-2o 8449 | . 2 ⊢ 2o = suc 1o | |
5 | 3, 4 | eqtr4i 2762 | 1 ⊢ (1o +o 1o) = 2o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Oncon0 6353 suc csuc 6355 (class class class)co 7393 1oc1o 8441 2oc2o 8442 +o coa 8445 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-oadd 8452 |
This theorem is referenced by: (None) |
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