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Mirrors > Home > ILE Home > Th. List > strle2g | GIF version |
Description: Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
strle2.j | ⊢ 𝐼 < 𝐽 |
strle2.k | ⊢ 𝐽 ∈ ℕ |
strle2.b | ⊢ 𝐵 = 𝐽 |
Ref | Expression |
---|---|
strle2g | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3534 | . 2 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} = ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) | |
2 | strle1.i | . . . . 5 ⊢ 𝐼 ∈ ℕ | |
3 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
4 | 2, 3 | strle1g 12059 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
6 | strle2.k | . . . . 5 ⊢ 𝐽 ∈ ℕ | |
7 | strle2.b | . . . . 5 ⊢ 𝐵 = 𝐽 | |
8 | 6, 7 | strle1g 12059 | . . . 4 ⊢ (𝑌 ∈ 𝑊 → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
10 | strle2.j | . . . 4 ⊢ 𝐼 < 𝐽 | |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝐼 < 𝐽) |
12 | 5, 9, 11 | strleund 12057 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) Struct 〈𝐼, 𝐽〉) |
13 | 1, 12 | eqbrtrid 3963 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∪ cun 3069 {csn 3527 {cpr 3528 〈cop 3530 class class class wbr 3929 < clt 7807 ℕcn 8727 Struct cstr 11965 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-n0 8985 df-z 9062 df-uz 9334 df-fz 9798 df-struct 11971 |
This theorem is referenced by: strle3g 12061 2strstrg 12069 |
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