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Mirrors > Home > ILE Home > Th. List > tpfidisj | GIF version |
Description: A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.) |
Ref | Expression |
---|---|
tpfidisj.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
tpfidisj.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
tpfidisj.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
tpfidisj.ab | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tpfidisj.ac | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
tpfidisj.bc | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
tpfidisj | ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3535 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | tpfidisj.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | tpfidisj.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | tpfidisj.ab | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
5 | prfidisj 6815 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3anc 1216 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
7 | tpfidisj.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | snfig 6708 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → {𝐶} ∈ Fin) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝜑 → {𝐶} ∈ Fin) |
10 | tpfidisj.ac | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 10 | necomd 2394 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
12 | tpfidisj.bc | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
13 | 12 | necomd 2394 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 11, 13 | nelprd 3553 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
15 | disjsn 3585 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
16 | 14, 15 | sylibr 133 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
17 | unfidisj 6810 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin ∧ ({𝐴, 𝐵} ∩ {𝐶}) = ∅) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
18 | 6, 9, 16, 17 | syl3anc 1216 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) |
19 | 1, 18 | eqeltrid 2226 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∪ cun 3069 ∩ cin 3070 ∅c0 3363 {csn 3527 {cpr 3528 {ctp 3529 Fincfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: sumtp 11183 |
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