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Theorem cfslb 9032
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 6158 . . 3 (card‘𝐵) ∈ V
2 ssid 3603 . . . . . . 7 (card‘𝐵) ⊆ (card‘𝐵)
3 cfslb.1 . . . . . . . . . . 11 𝐴 ∈ V
43ssex 4762 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
54ad2antrr 761 . . . . . . . . 9 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6 selpw 4137 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sseq1 3605 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
86, 7syl5bb 272 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴𝐵𝐴))
9 unieq 4410 . . . . . . . . . . . . 13 (𝑥 = 𝐵 𝑥 = 𝐵)
109eqeq1d 2623 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ( 𝑥 = 𝐴 𝐵 = 𝐴))
118, 10anbi12d 746 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝐵𝐴 𝐵 = 𝐴)))
12 fveq2 6148 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
1312sseq1d 3611 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
1411, 13anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
1514spcegv 3280 . . . . . . . . 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 2913 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18 rabid 3106 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
1918anbi1i 730 . . . . . . . . . 10 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2019exbii 1771 . . . . . . . . 9 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2117, 20bitri 264 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2216, 21sylibr 224 . . . . . . 7 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
232, 22mpan2 706 . . . . . 6 ((𝐵𝐴 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24 iinss 4537 . . . . . 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 9028 . . . . . 6 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
2726sseq1d 3611 . . . . 5 (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
2825, 27syl5ibr 236 . . . 4 (Lim 𝐴 → ((𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29283impib 1259 . . 3 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30 ssdomg 7945 . . 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))
324adantl 482 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ V)
33 limord 5743 . . . . . . 7 (Lim 𝐴 → Ord 𝐴)
34 ordsson 6936 . . . . . . 7 (Ord 𝐴𝐴 ⊆ On)
3533, 34syl 17 . . . . . 6 (Lim 𝐴𝐴 ⊆ On)
36 sstr2 3590 . . . . . 6 (𝐵𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
3735, 36mpan9 486 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ⊆ On)
38 onssnum 8807 . . . . 5 ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
3932, 37, 38syl2anc 692 . . . 4 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
40 cardid2 8723 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
4139, 40syl 17 . . 3 ((Lim 𝐴𝐵𝐴) → (card‘𝐵) ≈ 𝐵)
42413adant3 1079 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
43 domentr 7959 . 2 (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
4431, 42, 43syl2anc 692 1 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   ciin 4486   class class class wbr 4613  dom cdm 5074  Ord word 5681  Oncon0 5682  Lim wlim 5683  cfv 5847  cen 7896  cdom 7897  cardccrd 8705  cfccf 8707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-wrecs 7352  df-recs 7413  df-er 7687  df-en 7900  df-dom 7901  df-card 8709  df-cf 8711
This theorem is referenced by:  cfslbn  9033  cfslb2n  9034  rankcf  9543
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