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Theorem cfslb 10304
Description: Any cofinal subset of 𝐴 is at least as large as (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1 𝐴 ∈ V
Assertion
Ref Expression
cfslb ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Proof of Theorem cfslb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvex 6920 . . 3 (card‘𝐵) ∈ V
2 ssid 4018 . . . . . . 7 (card‘𝐵) ⊆ (card‘𝐵)
3 cfslb.1 . . . . . . . . . . 11 𝐴 ∈ V
43ssex 5327 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
54ad2antrr 726 . . . . . . . . 9 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6 velpw 4610 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sseq1 4021 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
86, 7bitrid 283 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴𝐵𝐴))
9 unieq 4923 . . . . . . . . . . . . 13 (𝑥 = 𝐵 𝑥 = 𝐵)
109eqeq1d 2737 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ( 𝑥 = 𝐴 𝐵 = 𝐴))
118, 10anbi12d 632 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝐵𝐴 𝐵 = 𝐴)))
12 fveq2 6907 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
1312sseq1d 4027 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
1411, 13anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
1514spcegv 3597 . . . . . . . . 9 (𝐵 ∈ V → (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵))))
165, 15mpcom 38 . . . . . . . 8 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
17 df-rex 3069 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18 rabid 3455 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
1918anbi1i 624 . . . . . . . . . 10 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2019exbii 1845 . . . . . . . . 9 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2117, 20bitri 275 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2216, 21sylibr 234 . . . . . . 7 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
232, 22mpan2 691 . . . . . 6 ((𝐵𝐴 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24 iinss 5061 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
2523, 24syl 17 . . . . 5 ((𝐵𝐴 𝐵 = 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
263cflim3 10300 . . . . . 6 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
2726sseq1d 4027 . . . . 5 (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
2825, 27imbitrrid 246 . . . 4 (Lim 𝐴 → ((𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29283impib 1115 . . 3 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30 ssdomg 9039 . . 3 ((card‘𝐵) ∈ V → ((cf‘𝐴) ⊆ (card‘𝐵) → (cf‘𝐴) ≼ (card‘𝐵)))
311, 29, 30mpsyl 68 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ (card‘𝐵))
32 limord 6446 . . . . . . 7 (Lim 𝐴 → Ord 𝐴)
33 ordsson 7802 . . . . . . 7 (Ord 𝐴𝐴 ⊆ On)
3432, 33syl 17 . . . . . 6 (Lim 𝐴𝐴 ⊆ On)
35 sstr2 4002 . . . . . 6 (𝐵𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
3634, 35mpan9 506 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ⊆ On)
37 onssnum 10078 . . . . 5 ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
384, 36, 37syl2an2 686 . . . 4 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
39 cardid2 9991 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
4038, 39syl 17 . . 3 ((Lim 𝐴𝐵𝐴) → (card‘𝐵) ≈ 𝐵)
41403adant3 1131 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
42 domentr 9052 . 2 (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
4331, 41, 42syl2anc 584 1 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   ciin 4997   class class class wbr 5148  dom cdm 5689  Ord word 6385  Oncon0 6386  Lim wlim 6387  cfv 6563  cen 8981  cdom 8982  cardccrd 9973  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-card 9977  df-cf 9979
This theorem is referenced by:  cfslbn  10305  cfslb2n  10306  rankcf  10815
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