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Mirrors > Home > MPE Home > Th. List > ipopos | Structured version Visualization version GIF version |
Description: The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
ipopos.i | ⊢ 𝐼 = (toInc‘𝐹) |
Ref | Expression |
---|---|
ipopos | ⊢ 𝐼 ∈ Poset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipopos.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐹) | |
2 | 1 | fvexi 6684 | . . . 4 ⊢ 𝐼 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐹 ∈ V → 𝐼 ∈ V) |
4 | 1 | ipobas 17765 | . . 3 ⊢ (𝐹 ∈ V → 𝐹 = (Base‘𝐼)) |
5 | eqidd 2822 | . . 3 ⊢ (𝐹 ∈ V → (le‘𝐼) = (le‘𝐼)) | |
6 | ssid 3989 | . . . 4 ⊢ 𝑎 ⊆ 𝑎 | |
7 | eqid 2821 | . . . . . 6 ⊢ (le‘𝐼) = (le‘𝐼) | |
8 | 1, 7 | ipole 17768 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹) → (𝑎(le‘𝐼)𝑎 ↔ 𝑎 ⊆ 𝑎)) |
9 | 8 | 3anidm23 1417 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹) → (𝑎(le‘𝐼)𝑎 ↔ 𝑎 ⊆ 𝑎)) |
10 | 6, 9 | mpbiri 260 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹) → 𝑎(le‘𝐼)𝑎) |
11 | 1, 7 | ipole 17768 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹) → (𝑎(le‘𝐼)𝑏 ↔ 𝑎 ⊆ 𝑏)) |
12 | 1, 7 | ipole 17768 | . . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹) → (𝑏(le‘𝐼)𝑎 ↔ 𝑏 ⊆ 𝑎)) |
13 | 12 | 3com23 1122 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹) → (𝑏(le‘𝐼)𝑎 ↔ 𝑏 ⊆ 𝑎)) |
14 | 11, 13 | anbi12d 632 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹) → ((𝑎(le‘𝐼)𝑏 ∧ 𝑏(le‘𝐼)𝑎) ↔ (𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎))) |
15 | simpl 485 | . . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) → 𝑎 ⊆ 𝑏) | |
16 | simpr 487 | . . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) → 𝑏 ⊆ 𝑎) | |
17 | 15, 16 | eqssd 3984 | . . . 4 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) → 𝑎 = 𝑏) |
18 | 14, 17 | syl6bi 255 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹) → ((𝑎(le‘𝐼)𝑏 ∧ 𝑏(le‘𝐼)𝑎) → 𝑎 = 𝑏)) |
19 | sstr 3975 | . . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑐) → 𝑎 ⊆ 𝑐) | |
20 | 19 | a1i 11 | . . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑐) → 𝑎 ⊆ 𝑐)) |
21 | 11 | 3adant3r3 1180 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → (𝑎(le‘𝐼)𝑏 ↔ 𝑎 ⊆ 𝑏)) |
22 | 1, 7 | ipole 17768 | . . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹) → (𝑏(le‘𝐼)𝑐 ↔ 𝑏 ⊆ 𝑐)) |
23 | 22 | 3adant3r1 1178 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → (𝑏(le‘𝐼)𝑐 ↔ 𝑏 ⊆ 𝑐)) |
24 | 21, 23 | anbi12d 632 | . . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → ((𝑎(le‘𝐼)𝑏 ∧ 𝑏(le‘𝐼)𝑐) ↔ (𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑐))) |
25 | 1, 7 | ipole 17768 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑎 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹) → (𝑎(le‘𝐼)𝑐 ↔ 𝑎 ⊆ 𝑐)) |
26 | 25 | 3adant3r2 1179 | . . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → (𝑎(le‘𝐼)𝑐 ↔ 𝑎 ⊆ 𝑐)) |
27 | 20, 24, 26 | 3imtr4d 296 | . . 3 ⊢ ((𝐹 ∈ V ∧ (𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ∧ 𝑐 ∈ 𝐹)) → ((𝑎(le‘𝐼)𝑏 ∧ 𝑏(le‘𝐼)𝑐) → 𝑎(le‘𝐼)𝑐)) |
28 | 3, 4, 5, 10, 18, 27 | isposd 17565 | . 2 ⊢ (𝐹 ∈ V → 𝐼 ∈ Poset) |
29 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐹 ∈ V → (toInc‘𝐹) = ∅) | |
30 | 1, 29 | syl5eq 2868 | . . 3 ⊢ (¬ 𝐹 ∈ V → 𝐼 = ∅) |
31 | 0pos 17564 | . . 3 ⊢ ∅ ∈ Poset | |
32 | 30, 31 | eqeltrdi 2921 | . 2 ⊢ (¬ 𝐹 ∈ V → 𝐼 ∈ Poset) |
33 | 28, 32 | pm2.61i 184 | 1 ⊢ 𝐼 ∈ Poset |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 lecple 16572 Posetcpo 17550 toInccipo 17761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-tset 16584 df-ple 16585 df-ocomp 16586 df-poset 17556 df-ipo 17762 |
This theorem is referenced by: isipodrs 17771 mrelatglb 17794 mrelatglb0 17795 mrelatlub 17796 mreclatBAD 17797 |
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