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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 3568 | . 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 6871 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3anc 1220 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
7 | tpfidisj.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | snfig 6759 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → {𝐶} ∈ Fin) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝜑 → {𝐶} ∈ Fin) |
10 | tpfidisj.ac | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 10 | necomd 2413 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
12 | tpfidisj.bc | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
13 | 12 | necomd 2413 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 11, 13 | nelprd 3586 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
15 | disjsn 3621 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
16 | 14, 15 | sylibr 133 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
17 | unfidisj 6866 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin ∧ ({𝐴, 𝐵} ∩ {𝐶}) = ∅) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
18 | 6, 9, 16, 17 | syl3anc 1220 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) |
19 | 1, 18 | eqeltrid 2244 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 {cpr 3561 {ctp 3562 Fincfn 6685 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-tp 3568 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-er 6480 df-en 6686 df-fin 6688 |
This theorem is referenced by: sumtp 11311 |
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