Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hashsdom | Structured version Visualization version GIF version |
Description: Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
Ref | Expression |
---|---|
hashsdom | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴 ≺ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 13009 | . . . 4 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0) | |
2 | hashcl 13009 | . . . 4 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0) | |
3 | nn0re 11178 | . . . . 5 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ) | |
4 | nn0re 11178 | . . . . 5 ⊢ ((#‘𝐵) ∈ ℕ0 → (#‘𝐵) ∈ ℝ) | |
5 | ltlen 10017 | . . . . 5 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((#‘𝐴) < (#‘𝐵) ↔ ((#‘𝐴) ≤ (#‘𝐵) ∧ (#‘𝐵) ≠ (#‘𝐴)))) | |
6 | 3, 4, 5 | syl2an 493 | . . . 4 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) < (#‘𝐵) ↔ ((#‘𝐴) ≤ (#‘𝐵) ∧ (#‘𝐵) ≠ (#‘𝐴)))) |
7 | 1, 2, 6 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ ((#‘𝐴) ≤ (#‘𝐵) ∧ (#‘𝐵) ≠ (#‘𝐴)))) |
8 | hashdom 13029 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵)) | |
9 | eqcom 2617 | . . . . . 6 ⊢ ((#‘𝐵) = (#‘𝐴) ↔ (#‘𝐴) = (#‘𝐵)) | |
10 | hashen 12997 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
11 | 9, 10 | syl5bb 271 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐵) = (#‘𝐴) ↔ 𝐴 ≈ 𝐵)) |
12 | 11 | necon3abid 2818 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐵) ≠ (#‘𝐴) ↔ ¬ 𝐴 ≈ 𝐵)) |
13 | 8, 12 | anbi12d 743 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (((#‘𝐴) ≤ (#‘𝐵) ∧ (#‘𝐵) ≠ (#‘𝐴)) ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))) |
14 | 7, 13 | bitrd 267 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))) |
15 | brsdom 7864 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
16 | 14, 15 | syl6bbr 277 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 ℝcr 9814 < clt 9953 ≤ cle 9954 ℕ0cn0 11169 #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: fzsdom2 13075 vdwlem12 15534 odcau 17842 pgpssslw 17852 pgpfaclem2 18304 ppiltx 24703 erdszelem10 30436 rp-isfinite6 36883 |
Copyright terms: Public domain | W3C validator |