Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > acexmid | Unicode version |
Description: The axiom of choice
implies excluded middle. Theorem 1.3 in [Bauer]
p. 483.
The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function provides a value when is inhabited (as opposed to non-empty as in some statements of the axiom of choice). Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). (Contributed by Jim Kingdon, 4-Aug-2019.) |
Ref | Expression |
---|---|
acexmid.choice |
Ref | Expression |
---|---|
acexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . . . . . . . . . . . . . 14 | |
2 | 1 | sb8eu 1954 | . . . . . . . . . . . . 13 |
3 | eleq12 2143 | . . . . . . . . . . . . . . . . . . . 20 | |
4 | 3 | ancoms 264 | . . . . . . . . . . . . . . . . . . 19 |
5 | 4 | 3adant3 958 | . . . . . . . . . . . . . . . . . 18 |
6 | eleq12 2143 | . . . . . . . . . . . . . . . . . . . . 21 | |
7 | 6 | 3ad2antl1 1100 | . . . . . . . . . . . . . . . . . . . 20 |
8 | eleq12 2143 | . . . . . . . . . . . . . . . . . . . . 21 | |
9 | 8 | 3ad2antl2 1101 | . . . . . . . . . . . . . . . . . . . 20 |
10 | 7, 9 | anbi12d 456 | . . . . . . . . . . . . . . . . . . 19 |
11 | simpl3 943 | . . . . . . . . . . . . . . . . . . 19 | |
12 | 10, 11 | cbvrexdva2 2582 | . . . . . . . . . . . . . . . . . 18 |
13 | 5, 12 | anbi12d 456 | . . . . . . . . . . . . . . . . 17 |
14 | 13 | 3com23 1144 | . . . . . . . . . . . . . . . 16 |
15 | 14 | 3expa 1138 | . . . . . . . . . . . . . . 15 |
16 | 15 | sbiedv 1712 | . . . . . . . . . . . . . 14 |
17 | 16 | eubidv 1949 | . . . . . . . . . . . . 13 |
18 | 2, 17 | syl5bb 190 | . . . . . . . . . . . 12 |
19 | df-reu 2355 | . . . . . . . . . . . 12 | |
20 | df-reu 2355 | . . . . . . . . . . . 12 | |
21 | 18, 19, 20 | 3bitr4g 221 | . . . . . . . . . . 11 |
22 | 21 | adantr 270 | . . . . . . . . . 10 |
23 | simpll 495 | . . . . . . . . . 10 | |
24 | 22, 23 | cbvraldva2 2581 | . . . . . . . . 9 |
25 | 24 | ancoms 264 | . . . . . . . 8 |
26 | 25 | adantll 459 | . . . . . . 7 |
27 | simpll 495 | . . . . . . 7 | |
28 | 26, 27 | cbvraldva2 2581 | . . . . . 6 |
29 | 28 | cbvexdva 1845 | . . . . 5 |
30 | 29 | cbvalv 1835 | . . . 4 |
31 | acexmid.choice | . . . 4 | |
32 | 30, 31 | mpgbir 1382 | . . 3 |
33 | 32 | spi 1469 | . 2 |
34 | 33 | acexmidlemv 5530 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 102 wb 103 wo 661 w3a 919 wal 1282 wex 1421 wsb 1685 weu 1941 wral 2348 wrex 2349 wreu 2350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iota 4887 df-riota 5488 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |