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Mirrors > Home > MPE Home > Th. List > cfslbn | Structured version Visualization version GIF version |
Description: Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 10022.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
Ref | Expression |
---|---|
cfslb.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
cfslbn | ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniss 4847 | . . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴) | |
2 | limuni 6326 | . . . . . . . . 9 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
3 | 2 | sseq2d 3953 | . . . . . . . 8 ⊢ (Lim 𝐴 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ ∪ 𝐴)) |
4 | 1, 3 | syl5ibr 245 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴)) |
5 | 4 | imp 407 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∪ 𝐵 ⊆ 𝐴) |
6 | limord 6325 | . . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴) | |
7 | ordsson 7633 | . . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 6, 7 | syl 17 | . . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On) |
9 | sstr2 3928 | . . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On)) | |
10 | 8, 9 | syl5com 31 | . . . . . . . . . 10 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → 𝐵 ⊆ On)) |
11 | ssorduni 7629 | . . . . . . . . . 10 ⊢ (𝐵 ⊆ On → Ord ∪ 𝐵) | |
12 | 10, 11 | syl6 35 | . . . . . . . . 9 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → Ord ∪ 𝐵)) |
13 | 12, 6 | jctird 527 | . . . . . . . 8 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → (Ord ∪ 𝐵 ∧ Ord 𝐴))) |
14 | ordsseleq 6295 | . . . . . . . 8 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴))) | |
15 | 13, 14 | syl6 35 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴)))) |
16 | 15 | imp 407 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴))) |
17 | 5, 16 | mpbid 231 | . . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴)) |
18 | 17 | ord 861 | . . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ∪ 𝐵 = 𝐴)) |
19 | cfslb.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
20 | 19 | cfslb 10022 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵) |
21 | domnsym 8886 | . . . . . 6 ⊢ ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴)) | |
22 | 20, 21 | syl 17 | . . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ¬ 𝐵 ≺ (cf‘𝐴)) |
23 | 22 | 3expia 1120 | . . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴))) |
24 | 18, 23 | syld 47 | . . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴))) |
25 | 24 | con4d 115 | . 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ∈ 𝐴)) |
26 | 25 | 3impia 1116 | 1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∪ cuni 4839 class class class wbr 5074 Ord word 6265 Oncon0 6266 Lim wlim 6267 ‘cfv 6433 ≼ cdom 8731 ≺ csdm 8732 cfccf 9695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-card 9697 df-cf 9699 |
This theorem is referenced by: cfslb2n 10024 |
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