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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 27741 | . 2 ⊢ ( L ‘ 1s ) = {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } | |
2 | bday1s 27715 | . . . . . 6 ⊢ ( bday ‘ 1s ) = 1o | |
3 | 2 | fveq2i 6887 | . . . . 5 ⊢ ( O ‘( bday ‘ 1s )) = ( O ‘1o) |
4 | old1 27753 | . . . . 5 ⊢ ( O ‘1o) = { 0s } | |
5 | 3, 4 | eqtri 2754 | . . . 4 ⊢ ( O ‘( bday ‘ 1s )) = { 0s } |
6 | 5 | rabeqi 3439 | . . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s } |
7 | breq1 5144 | . . . 4 ⊢ (𝑥 = 0s → (𝑥 <s 1s ↔ 0s <s 1s )) | |
8 | 7 | rabsnif 4722 | . . 3 ⊢ {𝑥 ∈ { 0s } ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅) |
9 | 6, 8 | eqtri 2754 | . 2 ⊢ {𝑥 ∈ ( O ‘( bday ‘ 1s )) ∣ 𝑥 <s 1s } = if( 0s <s 1s , { 0s }, ∅) |
10 | 0slt1s 27713 | . . 3 ⊢ 0s <s 1s | |
11 | 10 | iftruei 4530 | . 2 ⊢ if( 0s <s 1s , { 0s }, ∅) = { 0s } |
12 | 1, 9, 11 | 3eqtri 2758 | 1 ⊢ ( L ‘ 1s ) = { 0s } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {crab 3426 ∅c0 4317 ifcif 4523 {csn 4623 class class class wbr 5141 ‘cfv 6536 1oc1o 8457 <s cslt 27525 bday cbday 27526 0s c0s 27706 1s c1s 27707 O cold 27721 L cleft 27723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-1o 8464 df-2o 8465 df-no 27527 df-slt 27528 df-bday 27529 df-sle 27629 df-sslt 27665 df-scut 27667 df-0s 27708 df-1s 27709 df-made 27725 df-old 27726 df-left 27728 |
This theorem is referenced by: mulsrid 27964 |
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