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Mirrors > Home > MPE Home > Th. List > ttukeylem2 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 9934. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeylem.1 | ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) |
ttukeylem.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
ttukeylem.3 | ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) |
Ref | Expression |
---|---|
ttukeylem2 | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝐷 ⊆ 𝐶) | |
2 | sspwb 5334 | . . . . . 6 ⊢ (𝐷 ⊆ 𝐶 ↔ 𝒫 𝐷 ⊆ 𝒫 𝐶) | |
3 | 1, 2 | sylib 220 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶) |
4 | ssrin 4210 | . . . . 5 ⊢ (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin)) | |
5 | sstr2 3974 | . . . . 5 ⊢ ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴)) | |
6 | 3, 4, 5 | 3syl 18 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴)) |
7 | ttukeylem.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) | |
8 | ttukeylem.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
9 | ttukeylem.3 | . . . . . 6 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) | |
10 | 7, 8, 9 | ttukeylem1 9925 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) |
12 | 7, 8, 9 | ttukeylem1 9925 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴)) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴)) |
14 | 6, 11, 13 | 3imtr4d 296 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 → 𝐷 ∈ 𝐴)) |
15 | 14 | impancom 454 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ⊆ 𝐶 → 𝐷 ∈ 𝐴)) |
16 | 15 | impr 457 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∈ wcel 2110 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 –1-1-onto→wf1o 6349 ‘cfv 6350 Fincfn 8503 cardccrd 9358 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-1o 8096 df-en 8504 df-dom 8505 df-fin 8507 |
This theorem is referenced by: ttukeylem6 9930 ttukeylem7 9931 |
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