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Mirrors > Home > ILE Home > Th. List > 1stinl | Unicode version |
Description: The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
1stinl | inl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 6925 | . . . . 5 inl | |
2 | 1 | a1i 9 | . . . 4 inl |
3 | opeq2 3701 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | elex 2692 | . . . 4 | |
6 | 0ex 4050 | . . . . 5 | |
7 | opexg 4145 | . . . . 5 | |
8 | 6, 7 | mpan 420 | . . . 4 |
9 | 2, 4, 5, 8 | fvmptd 5495 | . . 3 inl |
10 | 9 | fveq2d 5418 | . 2 inl |
11 | op1stg 6041 | . . 3 | |
12 | 6, 11 | mpan 420 | . 2 |
13 | 10, 12 | eqtrd 2170 | 1 inl |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cvv 2681 c0 3358 cop 3525 cmpt 3984 cfv 5118 c1st 6029 inlcinl 6923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-1st 6031 df-inl 6925 |
This theorem is referenced by: djune 6956 updjudhcoinlf 6958 subctctexmid 13185 |
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