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Mirrors > Home > ILE Home > Th. List > filtinf | GIF version |
Description: The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
filtinf | ⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < (♯‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 10250 | . . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
2 | 1 | nn0red 8788 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ) |
3 | ltpnf 9312 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℝ → (♯‘𝐴) < +∞) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) < +∞) |
5 | 4 | adantr 271 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < +∞) |
6 | hashinfom 10247 | . . 3 ⊢ (ω ≼ 𝐵 → (♯‘𝐵) = +∞) | |
7 | 6 | adantl 272 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐵) = +∞) |
8 | 5, 7 | breqtrrd 3877 | 1 ⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < (♯‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 ωcom 4418 ‘cfv 5028 ≼ cdom 6510 Fincfn 6511 ℝcr 7410 +∞cpnf 7580 < clt 7583 ♯chash 10244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-recs 6084 df-frec 6170 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812 df-uz 9081 df-ihash 10245 |
This theorem is referenced by: (None) |
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