Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pm54.43 | Structured version Visualization version Unicode version |
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. "From
this proposition it
will follow, when arithmetical addition has been defined, that
1+1=2."
See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations.
This theorem states that two sets of cardinality 1 are disjoint iff
their union has cardinality 2.
Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8794), so that their means, in our notation, which is the same as by pm54.43lem 8825. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem pm110.643 8999 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.) |
Ref | Expression |
---|---|
pm54.43 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7567 | . . . . . . . 8 | |
2 | 1 | elexi 3213 | . . . . . . 7 |
3 | 2 | ensn1 8020 | . . . . . 6 |
4 | 3 | ensymi 8006 | . . . . 5 |
5 | entr 8008 | . . . . 5 | |
6 | 4, 5 | mpan2 707 | . . . 4 |
7 | 1 | onirri 5834 | . . . . . . 7 |
8 | disjsn 4246 | . . . . . . 7 | |
9 | 7, 8 | mpbir 221 | . . . . . 6 |
10 | unen 8040 | . . . . . 6 | |
11 | 9, 10 | mpanr2 720 | . . . . 5 |
12 | 11 | ex 450 | . . . 4 |
13 | 6, 12 | sylan2 491 | . . 3 |
14 | df-2o 7561 | . . . . 5 | |
15 | df-suc 5729 | . . . . 5 | |
16 | 14, 15 | eqtri 2644 | . . . 4 |
17 | 16 | breq2i 4661 | . . 3 |
18 | 13, 17 | syl6ibr 242 | . 2 |
19 | en1 8023 | . . 3 | |
20 | en1 8023 | . . 3 | |
21 | unidm 3756 | . . . . . . . . . . . . . 14 | |
22 | sneq 4187 | . . . . . . . . . . . . . . 15 | |
23 | 22 | uneq2d 3767 | . . . . . . . . . . . . . 14 |
24 | 21, 23 | syl5reqr 2671 | . . . . . . . . . . . . 13 |
25 | vex 3203 | . . . . . . . . . . . . . . 15 | |
26 | 25 | ensn1 8020 | . . . . . . . . . . . . . 14 |
27 | 1sdom2 8159 | . . . . . . . . . . . . . 14 | |
28 | ensdomtr 8096 | . . . . . . . . . . . . . 14 | |
29 | 26, 27, 28 | mp2an 708 | . . . . . . . . . . . . 13 |
30 | 24, 29 | syl6eqbr 4692 | . . . . . . . . . . . 12 |
31 | sdomnen 7984 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl 17 | . . . . . . . . . . 11 |
33 | 32 | necon2ai 2823 | . . . . . . . . . 10 |
34 | disjsn2 4247 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 17 | . . . . . . . . 9 |
36 | 35 | a1i 11 | . . . . . . . 8 |
37 | uneq12 3762 | . . . . . . . . 9 | |
38 | 37 | breq1d 4663 | . . . . . . . 8 |
39 | ineq12 3809 | . . . . . . . . 9 | |
40 | 39 | eqeq1d 2624 | . . . . . . . 8 |
41 | 36, 38, 40 | 3imtr4d 283 | . . . . . . 7 |
42 | 41 | ex 450 | . . . . . 6 |
43 | 42 | exlimdv 1861 | . . . . 5 |
44 | 43 | exlimiv 1858 | . . . 4 |
45 | 44 | imp 445 | . . 3 |
46 | 19, 20, 45 | syl2anb 496 | . 2 |
47 | 18, 46 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 cun 3572 cin 3573 c0 3915 csn 4177 class class class wbr 4653 con0 5723 csuc 5725 c1o 7553 c2o 7554 cen 7952 csdm 7954 |
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-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-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-br 4654 df-opab 4713 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-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-om 7066 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: pr2nelem 8827 pm110.643 8999 |
Copyright terms: Public domain | W3C validator |