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Theorem cfslb 8971
 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 6113 . . 3 (card‘𝐵) ∈ V
2 ssid 3587 . . . . . . 7 (card‘𝐵) ⊆ (card‘𝐵)
3 cfslb.1 . . . . . . . . . . 11 𝐴 ∈ V
43ssex 4730 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
54ad2antrr 758 . . . . . . . . 9 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6 selpw 4115 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sseq1 3589 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
86, 7syl5bb 271 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴𝐵𝐴))
9 unieq 4380 . . . . . . . . . . . . 13 (𝑥 = 𝐵 𝑥 = 𝐵)
109eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ( 𝑥 = 𝐴 𝐵 = 𝐴))
118, 10anbi12d 743 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝐵𝐴 𝐵 = 𝐴)))
12 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
1312sseq1d 3595 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
1411, 13anbi12d 743 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
1514spcegv 3267 . . . . . . . . 9 (𝐵 ∈ V → (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵))))
165, 15mpcom 37 . . . . . . . 8 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
17 df-rex 2902 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18 rabid 3095 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
1918anbi1i 727 . . . . . . . . . 10 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2019exbii 1764 . . . . . . . . 9 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2117, 20bitri 263 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2216, 21sylibr 223 . . . . . . 7 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
232, 22mpan2 703 . . . . . 6 ((𝐵𝐴 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24 iinss 4507 . . . . . 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 8967 . . . . . 6 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
2726sseq1d 3595 . . . . 5 (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
2825, 27syl5ibr 235 . . . 4 (Lim 𝐴 → ((𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29283impib 1254 . . 3 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30 ssdomg 7887 . . 3 ((card‘𝐵) ∈ V → ((cf‘𝐴) ⊆ (card‘𝐵) → (cf‘𝐴) ≼ (card‘𝐵)))
311, 29, 30mpsyl 66 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ (card‘𝐵))
324adantl 481 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ V)
33 limord 5701 . . . . . . 7 (Lim 𝐴 → Ord 𝐴)
34 ordsson 6881 . . . . . . 7 (Ord 𝐴𝐴 ⊆ On)
3533, 34syl 17 . . . . . 6 (Lim 𝐴𝐴 ⊆ On)
36 sstr2 3575 . . . . . 6 (𝐵𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
3735, 36mpan9 485 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ⊆ On)
38 onssnum 8746 . . . . 5 ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
3932, 37, 38syl2anc 691 . . . 4 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
40 cardid2 8662 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
4139, 40syl 17 . . 3 ((Lim 𝐴𝐵𝐴) → (card‘𝐵) ≈ 𝐵)
42413adant3 1074 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
43 domentr 7901 . 2 (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
4431, 42, 43syl2anc 691 1 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∩ ciin 4456   class class class wbr 4583  dom cdm 5038  Ord word 5639  Oncon0 5640  Lim wlim 5641  ‘cfv 5804   ≈ cen 7838   ≼ cdom 7839  cardccrd 8644  cfccf 8646 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-card 8648  df-cf 8650 This theorem is referenced by:  cfslbn  8972  cfslb2n  8973  rankcf  9478
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