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Mirrors > Home > ILE Home > Th. List > prfidisj | Unicode version |
Description: A pair is finite if it consists of two unequal sets. For the case where , see snfig 6759. For the cases where one or both is a proper class, see prprc1 3667, prprc2 3668, or prprc 3669. (Contributed by Jim Kingdon, 31-May-2022.) |
Ref | Expression |
---|---|
prfidisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3567 | . 2 | |
2 | snfig 6759 | . . 3 | |
3 | snfig 6759 | . . 3 | |
4 | disjsn2 3622 | . . 3 | |
5 | unfidisj 6866 | . . 3 | |
6 | 2, 3, 4, 5 | syl3an 1262 | . 2 |
7 | 1, 6 | eqeltrid 2244 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 963 wceq 1335 wcel 2128 wne 2327 cun 3100 cin 3101 c0 3394 csn 3560 cpr 3561 cfn 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-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: tpfidisj 6872 fiprsshashgt1 10691 sumpr 11310 |
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