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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 472 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≼ 𝐵) | |
2 | simpr 476 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
3 | reldom 7847 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5083 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
5 | 4 | adantr 480 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
6 | hashdom 13029 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ V) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵)) | |
7 | 2, 5, 6 | syl2anc 691 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵)) |
8 | 1, 7 | mpbird 246 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵)) |
9 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
10 | pnfge 11840 | . . . 4 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → +∞ ≤ +∞) |
12 | 3 | brrelexi 5082 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
13 | hashinf 12984 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) | |
14 | 12, 13 | sylan 487 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) |
15 | 4 | adantr 480 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
16 | domfi 8066 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵) → 𝐴 ∈ Fin) | |
17 | 16 | stoic1b 1689 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ∈ Fin) |
18 | hashinf 12984 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞) | |
19 | 15, 17, 18 | syl2anc 691 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐵) = +∞) |
20 | 11, 14, 19 | 3brtr4d 4615 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵)) |
21 | 8, 20 | pm2.61dan 828 | 1 ⊢ (𝐴 ≼ 𝐵 → (#‘𝐴) ≤ (#‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 ≼ cdom 7839 Fincfn 7841 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: hashge0 13037 o1fsum 14386 incexc2 14409 dchrisum0re 25002 usgraedgleord 25923 esumcst 29452 idomodle 36793 usgredgleord 40455 uspgredgaleord 40459 |
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