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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 10304.) (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 4920 | . . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴) | |
2 | limuni 6447 | . . . . . . . . 9 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
3 | 2 | sseq2d 4028 | . . . . . . . 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 6446 | . . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴) | |
7 | ordsson 7802 | . . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 6, 7 | syl 17 | . . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On) |
9 | sstr2 4002 | . . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On)) | |
10 | 8, 9 | syl5com 31 | . . . . . . . . . 10 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → 𝐵 ⊆ On)) |
11 | ssorduni 7798 | . . . . . . . . . 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 6415 | . . . . . . . 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 864 | . . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ∪ 𝐵 = 𝐴)) |
19 | cfslb.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
20 | 19 | cfslb 10304 | . . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵) |
21 | domnsym 9138 | . . . . . 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 Ord word 6385 Oncon0 6386 Lim wlim 6387 ‘cfv 6563 ≼ cdom 8982 ≺ csdm 8983 cfccf 9975 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-card 9977 df-cf 9979 |
This theorem is referenced by: cfslb2n 10306 |
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