Theorem cfslbn 9689

Description: Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 9688.) (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfslb.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cfslbn	⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴)

Proof of Theorem cfslbn

Step	Hyp	Ref	Expression
1		uniss 4846	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
2		limuni 6251	. . . . . . . . 9 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
3	2	sseq2d 3999	. . . . . . . 8 ⊢ (Lim 𝐴 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ ∪ 𝐴))
4	1, 3	syl5ibr 248	. . . . . . 7 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴))
5	4	imp 409	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∪ 𝐵 ⊆ 𝐴)
6		limord 6250	. . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴)
7		ordsson 7504	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
8	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
9		sstr2 3974	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
10	8, 9	syl5com 31	. . . . . . . . . 10 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → 𝐵 ⊆ On))
11		ssorduni 7500	. . . . . . . . . 10 ⊢ (𝐵 ⊆ On → Ord ∪ 𝐵)
12	10, 11	syl6 35	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → Ord ∪ 𝐵))
13	12, 6	jctird 529	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → (Ord ∪ 𝐵 ∧ Ord 𝐴)))
14		ordsseleq 6220	. . . . . . . 8 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴)))
15	13, 14	syl6 35	. . . . . . 7 ⊢ (Lim 𝐴 → (𝐵 ⊆ 𝐴 → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴))))
16	15	imp 409	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴)))
17	5, 16	mpbid 234	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 ∈ 𝐴 ∨ ∪ 𝐵 = 𝐴))
18	17	ord 860	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ∪ 𝐵 = 𝐴))
19		cfslb.1	. . . . . . 7 ⊢ 𝐴 ∈ V
20	19	cfslb 9688	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
21		domnsym 8643	. . . . . 6 ⊢ ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴))
22	20, 21	syl 17	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ¬ 𝐵 ≺ (cf‘𝐴))
23	22	3expia 1117	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
24	18, 23	syld 47	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∪ 𝐵 ∈ 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
25	24	con4d 115	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ∈ 𝐴))
26	25	3impia 1113	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  ∪ cuni 4838   class class class wbr 5066  Ord word 6190  Oncon0 6191  Lim wlim 6192  ‘cfv 6355   ≼ cdom 8507   ≺ csdm 8508  cfccf 9366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-wrecs 7947  df-recs 8008  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-card 9368  df-cf 9370

This theorem is referenced by:  cfslb2n  9690