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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 485 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≼ 𝐵) | |
2 | simpr 487 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
3 | reldom 8518 | . . . . . 6 ⊢ Rel ≼ | |
4 | 3 | brrelex2i 5612 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
6 | hashdom 13743 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ V) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵)) | |
7 | 2, 5, 6 | syl2anc 586 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵)) |
8 | 1, 7 | mpbird 259 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵)) |
9 | pnfxr 10698 | . . . 4 ⊢ +∞ ∈ ℝ* | |
10 | pnfge 12528 | . . . 4 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → +∞ ≤ +∞) |
12 | 3 | brrelex1i 5611 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
13 | hashinf 13698 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
14 | 12, 13 | sylan 582 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) |
15 | 4 | adantr 483 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → 𝐵 ∈ V) |
16 | domfi 8742 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵) → 𝐴 ∈ Fin) | |
17 | 16 | stoic1b 1773 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ∈ Fin) |
18 | hashinf 13698 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) | |
19 | 15, 17, 18 | syl2anc 586 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐵) = +∞) |
20 | 11, 14, 19 | 3brtr4d 5101 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵)) |
21 | 8, 20 | pm2.61dan 811 | 1 ⊢ (𝐴 ≼ 𝐵 → (♯‘𝐴) ≤ (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 ‘cfv 6358 ≼ cdom 8510 Fincfn 8512 +∞cpnf 10675 ℝ*cxr 10677 ≤ cle 10679 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: hashge0 13751 o1fsum 15171 incexc2 15196 usgriedgleord 27013 uspgredgleord 27017 esumcst 31326 idomodle 39802 |
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