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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 3591 | . 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 6904 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3anc 1233 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
7 | tpfidisj.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
8 | snfig 6792 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → {𝐶} ∈ Fin) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝜑 → {𝐶} ∈ Fin) |
10 | tpfidisj.ac | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 10 | necomd 2426 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
12 | tpfidisj.bc | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
13 | 12 | necomd 2426 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 11, 13 | nelprd 3609 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
15 | disjsn 3645 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
16 | 14, 15 | sylibr 133 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
17 | unfidisj 6899 | . . 3 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐶} ∈ Fin ∧ ({𝐴, 𝐵} ∩ {𝐶}) = ∅) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) | |
18 | 6, 9, 16, 17 | syl3anc 1233 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ∪ {𝐶}) ∈ Fin) |
19 | 1, 18 | eqeltrid 2257 | 1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∪ cun 3119 ∩ cin 3120 ∅c0 3414 {csn 3583 {cpr 3584 {ctp 3585 Fincfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-tp 3591 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-er 6513 df-en 6719 df-fin 6721 |
This theorem is referenced by: sumtp 11377 |
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