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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem1 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32576. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem1.a | ⊢ 𝐴 ⊆ 𝑋 |
kur14lem1.c | ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇 |
kur14lem1.k | ⊢ (𝐾‘𝐴) ∈ 𝑇 |
Ref | Expression |
---|---|
kur14lem1 | ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem1.a | . . 3 ⊢ 𝐴 ⊆ 𝑋 | |
2 | sseq1 3940 | . . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
3 | 1, 2 | mpbiri 261 | . 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋) |
4 | difeq2 4044 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴)) | |
5 | fveq2 6645 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴)) | |
6 | 4, 5 | preq12d 4637 | . . 3 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} = {(𝑋 ∖ 𝐴), (𝐾‘𝐴)}) |
7 | kur14lem1.c | . . . 4 ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇 | |
8 | kur14lem1.k | . . . 4 ⊢ (𝐾‘𝐴) ∈ 𝑇 | |
9 | prssi 4714 | . . . 4 ⊢ (((𝑋 ∖ 𝐴) ∈ 𝑇 ∧ (𝐾‘𝐴) ∈ 𝑇) → {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇) | |
10 | 7, 8, 9 | mp2an 691 | . . 3 ⊢ {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇 |
11 | 6, 10 | eqsstrdi 3969 | . 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇) |
12 | 3, 11 | jca 515 | 1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 {cpr 4527 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 |
This theorem is referenced by: kur14lem7 32572 |
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