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Theorem left1s 27948
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 27917 . 2 ( L ‘ 1s ) = {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s }
2 bday1s 27891 . . . . . 6 ( bday ‘ 1s ) = 1o
32fveq2i 6910 . . . . 5 ( O ‘( bday ‘ 1s )) = ( O ‘1o)
4 old1 27929 . . . . 5 ( O ‘1o) = { 0s }
53, 4eqtri 2763 . . . 4 ( O ‘( bday ‘ 1s )) = { 0s }
65rabeqi 3447 . . 3 {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s }
7 breq1 5151 . . . 4 (𝑥 = 0s → (𝑥 <s 1s ↔ 0s <s 1s ))
87rabsnif 4728 . . 3 {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅)
96, 8eqtri 2763 . 2 {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅)
10 0slt1s 27889 . . 3 0s <s 1s
1110iftruei 4538 . 2 if( 0s <s 1s , { 0s }, ∅) = { 0s }
121, 9, 113eqtri 2767 1 ( L ‘ 1s ) = { 0s }
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {crab 3433  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  cfv 6563  1oc1o 8498   <s cslt 27700   bday cbday 27701   0s c0s 27882   1s c1s 27883   O cold 27897   L cleft 27899
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904
This theorem is referenced by:  negs1s  28074  mulsrid  28154
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