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 nonempty 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). For this theorem stated using the df-ac 7030 and df-exmid 4089 syntaxes, see exmidac 7033. (Contributed by Jim Kingdon, 4-Aug-2019.) |
Ref | Expression |
---|---|
acexmid.choice |
Ref | Expression |
---|---|
acexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . . . . . . . . . . . . . 14 | |
2 | 1 | sb8eu 1990 | . . . . . . . . . . . . 13 |
3 | eleq12 2182 | . . . . . . . . . . . . . . . . . . . 20 | |
4 | 3 | ancoms 266 | . . . . . . . . . . . . . . . . . . 19 |
5 | 4 | 3adant3 986 | . . . . . . . . . . . . . . . . . 18 |
6 | eleq12 2182 | . . . . . . . . . . . . . . . . . . . . 21 | |
7 | 6 | 3ad2antl1 1128 | . . . . . . . . . . . . . . . . . . . 20 |
8 | eleq12 2182 | . . . . . . . . . . . . . . . . . . . . 21 | |
9 | 8 | 3ad2antl2 1129 | . . . . . . . . . . . . . . . . . . . 20 |
10 | 7, 9 | anbi12d 464 | . . . . . . . . . . . . . . . . . . 19 |
11 | simpl3 971 | . . . . . . . . . . . . . . . . . . 19 | |
12 | 10, 11 | cbvrexdva2 2636 | . . . . . . . . . . . . . . . . . 18 |
13 | 5, 12 | anbi12d 464 | . . . . . . . . . . . . . . . . 17 |
14 | 13 | 3com23 1172 | . . . . . . . . . . . . . . . 16 |
15 | 14 | 3expa 1166 | . . . . . . . . . . . . . . 15 |
16 | 15 | sbiedv 1747 | . . . . . . . . . . . . . 14 |
17 | 16 | eubidv 1985 | . . . . . . . . . . . . 13 |
18 | 2, 17 | syl5bb 191 | . . . . . . . . . . . 12 |
19 | df-reu 2400 | . . . . . . . . . . . 12 | |
20 | df-reu 2400 | . . . . . . . . . . . 12 | |
21 | 18, 19, 20 | 3bitr4g 222 | . . . . . . . . . . 11 |
22 | 21 | adantr 274 | . . . . . . . . . 10 |
23 | simpll 503 | . . . . . . . . . 10 | |
24 | 22, 23 | cbvraldva2 2635 | . . . . . . . . 9 |
25 | 24 | ancoms 266 | . . . . . . . 8 |
26 | 25 | adantll 467 | . . . . . . 7 |
27 | simpll 503 | . . . . . . 7 | |
28 | 26, 27 | cbvraldva2 2635 | . . . . . 6 |
29 | 28 | cbvexdva 1881 | . . . . 5 |
30 | 29 | cbvalv 1871 | . . . 4 |
31 | acexmid.choice | . . . 4 | |
32 | 30, 31 | mpgbir 1414 | . . 3 |
33 | 32 | spi 1501 | . 2 |
34 | 33 | acexmidlemv 5740 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 682 w3a 947 wal 1314 wex 1453 wsb 1720 weu 1977 wral 2393 wrex 2394 wreu 2395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iota 5058 df-riota 5698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |