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Theorem cfslb 10261
Description: Any cofinal subset of 𝐴 is at least as large as (cfβ€˜π΄). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1 𝐴 ∈ V
Assertion
Ref Expression
cfslb ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cfβ€˜π΄) β‰Ό 𝐡)

Proof of Theorem cfslb
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 fvex 6905 . . 3 (cardβ€˜π΅) ∈ V
2 ssid 4005 . . . . . . 7 (cardβ€˜π΅) βŠ† (cardβ€˜π΅)
3 cfslb.1 . . . . . . . . . . 11 𝐴 ∈ V
43ssex 5322 . . . . . . . . . 10 (𝐡 βŠ† 𝐴 β†’ 𝐡 ∈ V)
54ad2antrr 725 . . . . . . . . 9 (((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΅)) β†’ 𝐡 ∈ V)
6 velpw 4608 . . . . . . . . . . . . 13 (π‘₯ ∈ 𝒫 𝐴 ↔ π‘₯ βŠ† 𝐴)
7 sseq1 4008 . . . . . . . . . . . . 13 (π‘₯ = 𝐡 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝐡 βŠ† 𝐴))
86, 7bitrid 283 . . . . . . . . . . . 12 (π‘₯ = 𝐡 β†’ (π‘₯ ∈ 𝒫 𝐴 ↔ 𝐡 βŠ† 𝐴))
9 unieq 4920 . . . . . . . . . . . . 13 (π‘₯ = 𝐡 β†’ βˆͺ π‘₯ = βˆͺ 𝐡)
109eqeq1d 2735 . . . . . . . . . . . 12 (π‘₯ = 𝐡 β†’ (βˆͺ π‘₯ = 𝐴 ↔ βˆͺ 𝐡 = 𝐴))
118, 10anbi12d 632 . . . . . . . . . . 11 (π‘₯ = 𝐡 β†’ ((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ↔ (𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴)))
12 fveq2 6892 . . . . . . . . . . . 12 (π‘₯ = 𝐡 β†’ (cardβ€˜π‘₯) = (cardβ€˜π΅))
1312sseq1d 4014 . . . . . . . . . . 11 (π‘₯ = 𝐡 β†’ ((cardβ€˜π‘₯) βŠ† (cardβ€˜π΅) ↔ (cardβ€˜π΅) βŠ† (cardβ€˜π΅)))
1411, 13anbi12d 632 . . . . . . . . . 10 (π‘₯ = 𝐡 β†’ (((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)) ↔ ((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΅))))
1514spcegv 3588 . . . . . . . . 9 (𝐡 ∈ V β†’ (((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΅)) β†’ βˆƒπ‘₯((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅))))
165, 15mpcom 38 . . . . . . . 8 (((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΅)) β†’ βˆƒπ‘₯((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
17 df-rex 3072 . . . . . . . . 9 (βˆƒπ‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅) ↔ βˆƒπ‘₯(π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
18 rabid 3453 . . . . . . . . . . 11 (π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} ↔ (π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴))
1918anbi1i 625 . . . . . . . . . 10 ((π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)) ↔ ((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
2019exbii 1851 . . . . . . . . 9 (βˆƒπ‘₯(π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)) ↔ βˆƒπ‘₯((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
2117, 20bitri 275 . . . . . . . 8 (βˆƒπ‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅) ↔ βˆƒπ‘₯((π‘₯ ∈ 𝒫 𝐴 ∧ βˆͺ π‘₯ = 𝐴) ∧ (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
2216, 21sylibr 233 . . . . . . 7 (((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΅)) β†’ βˆƒπ‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅))
232, 22mpan2 690 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ βˆƒπ‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅))
24 iinss 5060 . . . . . 6 (βˆƒπ‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅) β†’ ∩ π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅))
2523, 24syl 17 . . . . 5 ((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ ∩ π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅))
263cflim3 10257 . . . . . 6 (Lim 𝐴 β†’ (cfβ€˜π΄) = ∩ π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯))
2726sseq1d 4014 . . . . 5 (Lim 𝐴 β†’ ((cfβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ ∩ π‘₯ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ βˆͺ π‘₯ = 𝐴} (cardβ€˜π‘₯) βŠ† (cardβ€˜π΅)))
2825, 27imbitrrid 245 . . . 4 (Lim 𝐴 β†’ ((𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cfβ€˜π΄) βŠ† (cardβ€˜π΅)))
29283impib 1117 . . 3 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cfβ€˜π΄) βŠ† (cardβ€˜π΅))
30 ssdomg 8996 . . 3 ((cardβ€˜π΅) ∈ V β†’ ((cfβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ (cfβ€˜π΄) β‰Ό (cardβ€˜π΅)))
311, 29, 30mpsyl 68 . 2 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cfβ€˜π΄) β‰Ό (cardβ€˜π΅))
32 limord 6425 . . . . . . 7 (Lim 𝐴 β†’ Ord 𝐴)
33 ordsson 7770 . . . . . . 7 (Ord 𝐴 β†’ 𝐴 βŠ† On)
3432, 33syl 17 . . . . . 6 (Lim 𝐴 β†’ 𝐴 βŠ† On)
35 sstr2 3990 . . . . . 6 (𝐡 βŠ† 𝐴 β†’ (𝐴 βŠ† On β†’ 𝐡 βŠ† On))
3634, 35mpan9 508 . . . . 5 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴) β†’ 𝐡 βŠ† On)
37 onssnum 10035 . . . . 5 ((𝐡 ∈ V ∧ 𝐡 βŠ† On) β†’ 𝐡 ∈ dom card)
384, 36, 37syl2an2 685 . . . 4 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴) β†’ 𝐡 ∈ dom card)
39 cardid2 9948 . . . 4 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
4038, 39syl 17 . . 3 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
41403adant3 1133 . 2 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
42 domentr 9009 . 2 (((cfβ€˜π΄) β‰Ό (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ (cfβ€˜π΄) β‰Ό 𝐡)
4331, 41, 42syl2anc 585 1 ((Lim 𝐴 ∧ 𝐡 βŠ† 𝐴 ∧ βˆͺ 𝐡 = 𝐴) β†’ (cfβ€˜π΄) β‰Ό 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  βˆ© ciin 4999   class class class wbr 5149  dom cdm 5677  Ord word 6364  Oncon0 6365  Lim wlim 6366  β€˜cfv 6544   β‰ˆ cen 8936   β‰Ό cdom 8937  cardccrd 9930  cfccf 9932
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-en 8940  df-dom 8941  df-card 9934  df-cf 9936
This theorem is referenced by:  cfslbn  10262  cfslb2n  10263  rankcf  10772
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