MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfslbn Structured version   Visualization version   GIF version

Theorem cfslbn 9681
Description: Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 9680.) (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1 𝐴 ∈ V
Assertion
Ref Expression
cfslbn ((Lim 𝐴𝐵𝐴𝐵 ≺ (cf‘𝐴)) → 𝐵𝐴)

Proof of Theorem cfslbn
StepHypRef Expression
1 uniss 4851 . . . . . . . 8 (𝐵𝐴 𝐵 𝐴)
2 limuni 6244 . . . . . . . . 9 (Lim 𝐴𝐴 = 𝐴)
32sseq2d 3997 . . . . . . . 8 (Lim 𝐴 → ( 𝐵𝐴 𝐵 𝐴))
41, 3syl5ibr 248 . . . . . . 7 (Lim 𝐴 → (𝐵𝐴 𝐵𝐴))
54imp 409 . . . . . 6 ((Lim 𝐴𝐵𝐴) → 𝐵𝐴)
6 limord 6243 . . . . . . . . . . . 12 (Lim 𝐴 → Ord 𝐴)
7 ordsson 7496 . . . . . . . . . . . 12 (Ord 𝐴𝐴 ⊆ On)
86, 7syl 17 . . . . . . . . . . 11 (Lim 𝐴𝐴 ⊆ On)
9 sstr2 3972 . . . . . . . . . . 11 (𝐵𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
108, 9syl5com 31 . . . . . . . . . 10 (Lim 𝐴 → (𝐵𝐴𝐵 ⊆ On))
11 ssorduni 7492 . . . . . . . . . 10 (𝐵 ⊆ On → Ord 𝐵)
1210, 11syl6 35 . . . . . . . . 9 (Lim 𝐴 → (𝐵𝐴 → Ord 𝐵))
1312, 6jctird 529 . . . . . . . 8 (Lim 𝐴 → (𝐵𝐴 → (Ord 𝐵 ∧ Ord 𝐴)))
14 ordsseleq 6213 . . . . . . . 8 ((Ord 𝐵 ∧ Ord 𝐴) → ( 𝐵𝐴 ↔ ( 𝐵𝐴 𝐵 = 𝐴)))
1513, 14syl6 35 . . . . . . 7 (Lim 𝐴 → (𝐵𝐴 → ( 𝐵𝐴 ↔ ( 𝐵𝐴 𝐵 = 𝐴))))
1615imp 409 . . . . . 6 ((Lim 𝐴𝐵𝐴) → ( 𝐵𝐴 ↔ ( 𝐵𝐴 𝐵 = 𝐴)))
175, 16mpbid 234 . . . . 5 ((Lim 𝐴𝐵𝐴) → ( 𝐵𝐴 𝐵 = 𝐴))
1817ord 860 . . . 4 ((Lim 𝐴𝐵𝐴) → (¬ 𝐵𝐴 𝐵 = 𝐴))
19 cfslb.1 . . . . . . 7 𝐴 ∈ V
2019cfslb 9680 . . . . . 6 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
21 domnsym 8635 . . . . . 6 ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴))
2220, 21syl 17 . . . . 5 ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → ¬ 𝐵 ≺ (cf‘𝐴))
23223expia 1115 . . . 4 ((Lim 𝐴𝐵𝐴) → ( 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
2418, 23syld 47 . . 3 ((Lim 𝐴𝐵𝐴) → (¬ 𝐵𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
2524con4d 115 . 2 ((Lim 𝐴𝐵𝐴) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
26253impia 1111 1 ((Lim 𝐴𝐵𝐴𝐵 ≺ (cf‘𝐴)) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3493  wss 3934   cuni 4830   class class class wbr 5057  Ord word 6183  Oncon0 6184  Lim wlim 6185  cfv 6348  cdom 8499  csdm 8500  cfccf 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-wrecs 7939  df-recs 8000  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-card 9360  df-cf 9362
This theorem is referenced by:  cfslb2n  9682
  Copyright terms: Public domain W3C validator