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Theorem odubas 17737

Description: Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)

**Hypotheses**
Ref	Expression
oduval.d	⊢ 𝐷 = (ODual‘𝑂)
odubas.b	⊢ 𝐵 = (Base‘𝑂)

**Assertion**
Ref	Expression
odubas	⊢ 𝐵 = (Base‘𝐷)

**Proof of Theorem odubas**
Step	Hyp	Ref	Expression
1		baseid 16537	. . 3 ⊢ Base = Slot (Base‘ndx)
2		1re 10635	. . . . 5 ⊢ 1 ∈ ℝ
3		1lt10 12231	. . . . 5 ⊢ 1 < ;10
4	2, 3	ltneii 10747	. . . 4 ⊢ 1 ≠ ;10
5		basendx 16541	. . . . 5 ⊢ (Base‘ndx) = 1
6		plendx 16660	. . . . 5 ⊢ (le‘ndx) = ;10
7	5, 6	neeq12i 3082	. . . 4 ⊢ ((Base‘ndx) ≠ (le‘ndx) ↔ 1 ≠ ;10)
8	4, 7	mpbir 233	. . 3 ⊢ (Base‘ndx) ≠ (le‘ndx)
9	1, 8	setsnid 16533	. 2 ⊢ (Base‘𝑂) = (Base‘(𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩))
10		odubas.b	. 2 ⊢ 𝐵 = (Base‘𝑂)
11		oduval.d	. . . 4 ⊢ 𝐷 = (ODual‘𝑂)
12		eqid 2821	. . . 4 ⊢ (le‘𝑂) = (le‘𝑂)
13	11, 12	oduval 17734	. . 3 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩)
14	13	fveq2i 6667	. 2 ⊢ (Base‘𝐷) = (Base‘(𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩))
15	9, 10, 14	3eqtr4i 2854	1 ⊢ 𝐵 = (Base‘𝐷)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ≠ wne 3016 ⟨cop 4566 ^◡ccnv 5548 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 cdc 12092 ndxcnx 16474 sSet csts 16475 Basecbs 16477 lecple 16566 ODualcodu 17732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ple 16579 df-odu 17733

This theorem is referenced by: odupos 17739 oduposb 17740 oduglb 17743 odulub 17745 odulatb 17747 oduclatb 17748 posglbd 17754 latmass 17792 latdisd 17794 odudlatb 17800 dlatjmdi 17801 oduprs 30638 odutos 30645 ordtcnvNEW 31158 ordtrest2NEW 31161

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