| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 10306.) (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 4915 | . . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴) | |
| 2 | limuni 6445 | . . . . . . . . 9 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
| 3 | 2 | sseq2d 4016 | . . . . . . . 8 ⊢ (Lim 𝐴 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ ∪ 𝐴)) |
| 4 | 1, 3 | imbitrrid 246 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴)) |
| 5 | 4 | imp 406 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∪ 𝐵 ⊆ 𝐴) |
| 6 | limord 6444 | . . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 7 | ordsson 7803 | . . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
| 8 | 6, 7 | syl 17 | . . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On) |
| 9 | sstr2 3990 | . . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On)) | |
| 10 | 8, 9 | syl5com 31 | . . . . . . . . . 10 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → 𝐵 ⊆ On)) |
| 11 | ssorduni 7799 | . . . . . . . . . 10 ⊢ (𝐵 ⊆ On → Ord ∪ 𝐵) | |
| 12 | 10, 11 | syl6 35 | . . . . . . . . 9 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → Ord ∪ 𝐵)) |
| 13 | 12, 6 | jctird 526 | . . . . . . . 8 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → (Ord ∪ 𝐵 ∧ Ord 𝐴))) |
| 14 | ordsseleq 6413 | . . . . . . . 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 406 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴))) |
| 17 | 5, 16 | mpbid 232 | . . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴)) |
| 18 | 17 | ord 865 | . . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ∪ 𝐵 = 𝐴)) |
| 19 | cfslb.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
| 20 | 19 | cfslb 10306 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵) |
| 21 | domnsym 9139 | . . . . . 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 1122 | . . . 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 1118 | 1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 Ord word 6383 Oncon0 6384 Lim wlim 6385 ‘cfv 6561 ≼ cdom 8983 ≺ csdm 8984 cfccf 9977 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-card 9979 df-cf 9981 |
| This theorem is referenced by: cfslb2n 10308 |
| Copyright terms: Public domain | W3C validator |