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Mirrors > Home > ILE Home > Th. List > exmidmotap | GIF version |
Description: The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.) |
Ref | Expression |
---|---|
exmidmotap | ⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 529 | . . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 TAp 𝑥) | |
2 | exmidapne 7320 | . . . . . . . . 9 ⊢ (EXMID → (𝑟 TAp 𝑥 ↔ 𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})) | |
3 | 2 | adantr 276 | . . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → (𝑟 TAp 𝑥 ↔ 𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})) |
4 | 1, 3 | mpbid 147 | . . . . . . 7 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}) |
5 | simprr 531 | . . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 TAp 𝑥) | |
6 | exmidapne 7320 | . . . . . . . . 9 ⊢ (EXMID → (𝑠 TAp 𝑥 ↔ 𝑠 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})) | |
7 | 6 | adantr 276 | . . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → (𝑠 TAp 𝑥 ↔ 𝑠 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})) |
8 | 5, 7 | mpbid 147 | . . . . . . 7 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}) |
9 | 4, 8 | eqtr4d 2229 | . . . . . 6 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 = 𝑠) |
10 | 9 | ex 115 | . . . . 5 ⊢ (EXMID → ((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠)) |
11 | 10 | alrimivv 1886 | . . . 4 ⊢ (EXMID → ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠)) |
12 | tapeq1 7312 | . . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 TAp 𝑥 ↔ 𝑠 TAp 𝑥)) | |
13 | 12 | mo4 2103 | . . . 4 ⊢ (∃*𝑟 𝑟 TAp 𝑥 ↔ ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠)) |
14 | 11, 13 | sylibr 134 | . . 3 ⊢ (EXMID → ∃*𝑟 𝑟 TAp 𝑥) |
15 | 14 | alrimiv 1885 | . 2 ⊢ (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥) |
16 | 2onn 6574 | . . . 4 ⊢ 2o ∈ ω | |
17 | tapeq2 7313 | . . . . . 6 ⊢ (𝑥 = 2o → (𝑟 TAp 𝑥 ↔ 𝑟 TAp 2o)) | |
18 | 17 | mobidv 2078 | . . . . 5 ⊢ (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o)) |
19 | 18 | spcgv 2847 | . . . 4 ⊢ (2o ∈ ω → (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o)) |
20 | 16, 19 | ax-mp 5 | . . 3 ⊢ (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o) |
21 | 2omotap 7319 | . . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID) | |
22 | 20, 21 | syl 14 | . 2 ⊢ (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → EXMID) |
23 | 15, 22 | impbii 126 | 1 ⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 = wceq 1364 ∃*wmo 2043 ∈ wcel 2164 ≠ wne 2364 {copab 4089 EXMIDwem 4223 ωcom 4622 2oc2o 6463 TAp wtap 7309 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-exmid 4224 df-id 4324 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fo 5260 df-fv 5262 df-1st 6193 df-2nd 6194 df-1o 6469 df-2o 6470 df-pap 7308 df-tap 7310 |
This theorem is referenced by: (None) |
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