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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 3540 | . 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 6823 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3anc 1217 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
7 | tpfidisj.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | snfig 6716 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → {𝐶} ∈ Fin) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝜑 → {𝐶} ∈ Fin) |
10 | tpfidisj.ac | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 10 | necomd 2395 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
12 | tpfidisj.bc | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
13 | 12 | necomd 2395 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 11, 13 | nelprd 3558 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
15 | disjsn 3593 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
16 | 14, 15 | sylibr 133 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
17 | unfidisj 6818 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin ∧ ({𝐴, 𝐵} ∩ {𝐶}) = ∅) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
18 | 6, 9, 16, 17 | syl3anc 1217 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) |
19 | 1, 18 | eqeltrid 2227 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 {csn 3532 {cpr 3533 {ctp 3534 Fincfn 6642 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1o 6321 df-er 6437 df-en 6643 df-fin 6645 |
This theorem is referenced by: sumtp 11215 |
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