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Mirrors > Home > ILE Home > Th. List > prfidisj | GIF version |
Description: A pair is finite if it consists of two unequal sets. For the case where 𝐴 = 𝐵, see snfig 6676. For the cases where one or both is a proper class, see prprc1 3601, prprc2 3602, or prprc 3603. (Contributed by Jim Kingdon, 31-May-2022.) |
Ref | Expression |
---|---|
prfidisj | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3504 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfig 6676 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) | |
3 | snfig 6676 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ Fin) | |
4 | disjsn2 3556 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
5 | unfidisj 6778 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin ∧ ({𝐴} ∩ {𝐵}) = ∅) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3an 1243 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴} ∪ {𝐵}) ∈ Fin) |
7 | 1, 6 | eqeltrid 2204 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∪ cun 3039 ∩ cin 3040 ∅c0 3333 {csn 3497 {cpr 3498 Fincfn 6602 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: tpfidisj 6784 fiprsshashgt1 10531 sumpr 11150 |
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