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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 6841. For the cases where one or both is a proper class, see prprc1 3715, prprc2 3716, or prprc 3717. (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 3614 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snfig 6841 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) | |
3 | snfig 6841 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ Fin) | |
4 | disjsn2 3670 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
5 | unfidisj 6951 | . . 3 ⊢ (({𝐴} ∈ Fin ∧ {𝐵} ∈ Fin ∧ ({𝐴} ∩ {𝐵}) = ∅) → ({𝐴} ∪ {𝐵}) ∈ Fin) | |
6 | 2, 3, 4, 5 | syl3an 1291 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴} ∪ {𝐵}) ∈ Fin) |
7 | 1, 6 | eqeltrid 2276 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∪ cun 3142 ∩ cin 3143 ∅c0 3437 {csn 3607 {cpr 3608 Fincfn 6767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1o 6442 df-er 6560 df-en 6768 df-fin 6770 |
This theorem is referenced by: tpfidisj 6957 fiprsshashgt1 10832 sumpr 11456 |
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