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Mirrors > Home > MPE Home > Th. List > hashdomi | Structured version Visualization version GIF version |
Description: Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
hashdomi | ⊢ (𝐴 ≼ 𝐵 → (#‘𝐴) ≤ (#‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≼ 𝐵) | |
2 | simpr 477 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
3 | reldom 7961 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5159 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
6 | hashdom 13168 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ V) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵)) | |
7 | 2, 5, 6 | syl2anc 693 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵)) |
8 | 1, 7 | mpbird 247 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵)) |
9 | pnfxr 10092 | . . . 4 ⊢ +∞ ∈ ℝ* | |
10 | pnfge 11964 | . . . 4 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → +∞ ≤ +∞) |
12 | 3 | brrelexi 5158 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
13 | hashinf 13122 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
14 | 12, 13 | sylan 488 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) |
15 | 4 | adantr 481 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
16 | domfi 8181 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵) → 𝐴 ∈ Fin) | |
17 | 16 | stoic1b 1698 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ∈ Fin) |
18 | hashinf 13122 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞) | |
19 | 15, 17, 18 | syl2anc 693 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐵) = +∞) |
20 | 11, 14, 19 | 3brtr4d 4685 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵)) |
21 | 8, 20 | pm2.61dan 832 | 1 ⊢ (𝐴 ≼ 𝐵 → (#‘𝐴) ≤ (#‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 ≼ cdom 7953 Fincfn 7955 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: hashge0 13176 o1fsum 14545 incexc2 14570 dchrisum0re 25202 usgriedgleord 26120 uspgredgleord 26124 esumcst 30125 idomodle 37774 |
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