![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > exmidonfin | GIF version |
Description: If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6867 and nnon 4607. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
Ref | Expression |
---|---|
exmidonfin | ⊢ (ω = (On ∩ Fin) → EXMID) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . 4 ⊢ {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} = {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} | |
2 | 1 | exmidonfinlem 7187 | . . 3 ⊢ (ω = (On ∩ Fin) → DECID 𝑧 = {∅}) |
3 | 2 | adantr 276 | . 2 ⊢ ((ω = (On ∩ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅}) |
4 | 3 | exmid1dc 4198 | 1 ⊢ (ω = (On ∩ Fin) → EXMID) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 834 = wceq 1353 {crab 2459 ∩ cin 3128 ⊆ wss 3129 ∅c0 3422 {csn 3592 {cpr 3593 EXMIDwem 4192 Oncon0 4361 ωcom 4587 Fincfn 6735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-iinf 4585 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-br 4002 df-opab 4063 df-tr 4100 df-exmid 4193 df-id 4291 df-iord 4364 df-on 4366 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-1o 6412 df-2o 6413 df-er 6530 df-en 6736 df-fin 6738 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |