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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 6771. For the cases where one or both is a proper class, see prprc1 3678, prprc2 3679, or prprc 3680. (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 3577 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfig 6771 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) | |
3 | snfig 6771 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ Fin) | |
4 | disjsn2 3633 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
5 | unfidisj 6878 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin ∧ ({𝐴} ∩ {𝐵}) = ∅) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3an 1269 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴} ∪ {𝐵}) ∈ Fin) |
7 | 1, 6 | eqeltrid 2251 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∪ cun 3109 ∩ cin 3110 ∅c0 3404 {csn 3570 {cpr 3571 Fincfn 6697 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1o 6375 df-er 6492 df-en 6698 df-fin 6700 |
This theorem is referenced by: tpfidisj 6884 fiprsshashgt1 10719 sumpr 11340 |
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