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Theorem cfslb 9410
 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 6450 . . 3 (card‘𝐵) ∈ V
2 ssid 3848 . . . . . . 7 (card‘𝐵) ⊆ (card‘𝐵)
3 cfslb.1 . . . . . . . . . . 11 𝐴 ∈ V
43ssex 5029 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
54ad2antrr 717 . . . . . . . . 9 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6 selpw 4387 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sseq1 3851 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
86, 7syl5bb 275 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴𝐵𝐴))
9 unieq 4668 . . . . . . . . . . . . 13 (𝑥 = 𝐵 𝑥 = 𝐵)
109eqeq1d 2827 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ( 𝑥 = 𝐴 𝐵 = 𝐴))
118, 10anbi12d 624 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝐵𝐴 𝐵 = 𝐴)))
12 fveq2 6437 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
1312sseq1d 3857 . . . . . . . . . . 11 (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
1411, 13anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
1514spcegv 3511 . . . . . . . . 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 3123 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18 rabid 3326 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
1918anbi1i 617 . . . . . . . . . 10 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2019exbii 1947 . . . . . . . . 9 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2117, 20bitri 267 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
2216, 21sylibr 226 . . . . . . 7 (((𝐵𝐴 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
232, 22mpan2 682 . . . . . 6 ((𝐵𝐴 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24 iinss 4793 . . . . . 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 9406 . . . . . 6 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
2726sseq1d 3857 . . . . 5 (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
2825, 27syl5ibr 238 . . . 4 (Lim 𝐴 → ((𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29283impib 1148 . . 3 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30 ssdomg 8274 . . 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 475 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ V)
33 limord 6026 . . . . . . 7 (Lim 𝐴 → Ord 𝐴)
34 ordsson 7255 . . . . . . 7 (Ord 𝐴𝐴 ⊆ On)
3533, 34syl 17 . . . . . 6 (Lim 𝐴𝐴 ⊆ On)
36 sstr2 3834 . . . . . 6 (𝐵𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
3735, 36mpan9 502 . . . . 5 ((Lim 𝐴𝐵𝐴) → 𝐵 ⊆ On)
38 onssnum 9183 . . . . 5 ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
3932, 37, 38syl2anc 579 . . . 4 ((Lim 𝐴𝐵𝐴) → 𝐵 ∈ dom card)
40 cardid2 9099 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
4139, 40syl 17 . . 3 ((Lim 𝐴𝐵𝐴) → (card‘𝐵) ≈ 𝐵)
42413adant3 1166 . 2 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
43 domentr 8287 . 2 (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
4431, 42, 43syl2anc 579 1 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∃wrex 3118  {crab 3121  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4380  ∪ cuni 4660  ∩ ciin 4743   class class class wbr 4875  dom cdm 5346  Ord word 5966  Oncon0 5967  Lim wlim 5968  ‘cfv 6127   ≈ cen 8225   ≼ cdom 8226  cardccrd 9081  cfccf 9083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-wrecs 7677  df-recs 7739  df-er 8014  df-en 8229  df-dom 8230  df-card 9085  df-cf 9087 This theorem is referenced by:  cfslbn  9411  cfslb2n  9412  rankcf  9921
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