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Mirrors > Home > MPE Home > Th. List > left1s | Structured version Visualization version GIF version |
Description: The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.) |
Ref | Expression |
---|---|
left1s | ⊢ ( L ‘ 1s ) = { 0s } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leftval 27917 | . 2 ⊢ ( L ‘ 1s ) = {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } | |
2 | bday1s 27891 | . . . . . 6 ⊢ ( bday ‘ 1s ) = 1o | |
3 | 2 | fveq2i 6910 | . . . . 5 ⊢ ( O ‘( bday ‘ 1s )) = ( O ‘1o) |
4 | old1 27929 | . . . . 5 ⊢ ( O ‘1o) = { 0s } | |
5 | 3, 4 | eqtri 2763 | . . . 4 ⊢ ( O ‘( bday ‘ 1s )) = { 0s } |
6 | 5 | rabeqi 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 )) | |
8 | 7 | rabsnif 4728 | . . 3 ⊢ {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅) |
9 | 6, 8 | eqtri 2763 | . 2 ⊢ {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅) |
10 | 0slt1s 27889 | . . 3 ⊢ 0s <s 1s | |
11 | 10 | iftruei 4538 | . 2 ⊢ if( 0s <s 1s , { 0s }, ∅) = { 0s } |
12 | 1, 9, 11 | 3eqtri 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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