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Theorem left1s 27933
Description: The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
Assertion
Ref Expression
left1s ( L ‘ 1s ) = { 0s }

Proof of Theorem left1s
StepHypRef Expression
1 leftval 27902 . 2 ( L ‘ 1s ) = {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s }
2 bday1s 27876 . . . . . 6 ( bday ‘ 1s ) = 1o
32fveq2i 6909 . . . . 5 ( O ‘( bday ‘ 1s )) = ( O ‘1o)
4 old1 27914 . . . . 5 ( O ‘1o) = { 0s }
53, 4eqtri 2765 . . . 4 ( O ‘( bday ‘ 1s )) = { 0s }
65rabeqi 3450 . . 3 {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s }
7 breq1 5146 . . . 4 (𝑥 = 0s → (𝑥 <s 1s ↔ 0s <s 1s ))
87rabsnif 4723 . . 3 {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅)
96, 8eqtri 2765 . 2 {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅)
10 0slt1s 27874 . . 3 0s <s 1s
1110iftruei 4532 . 2 if( 0s <s 1s , { 0s }, ∅) = { 0s }
121, 9, 113eqtri 2769 1 ( L ‘ 1s ) = { 0s }
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {crab 3436  c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143  cfv 6561  1oc1o 8499   <s cslt 27685   bday cbday 27686   0s c0s 27867   1s c1s 27868   O cold 27882   L cleft 27884
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889
This theorem is referenced by:  negs1s  28059  mulsrid  28139
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