MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfslb2n Structured version   Visualization version   GIF version
Theorem cfslb2n 10259
Description: Any small collection of small subsets of 𝐴 cannot have union 𝐴, where "small" means smaller than the cofinality. This is a stronger version of cfslb 10257. This is a common application of cofinality: under AC, (β„΅β€˜1) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1 𝐴 ∈ V
Assertion
Ref Expression
cfslb2n ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (𝐡 β‰Ί (cfβ€˜π΄) β†’ βˆͺ 𝐡 β‰  𝐴))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Proof of Theorem cfslb2n
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limord 6421 . . . . . . . . . 10 (Lim 𝐴 β†’ Ord 𝐴)
2 ordsson 7766 . . . . . . . . . 10 (Ord 𝐴 β†’ 𝐴 βŠ† On)
3 sstr 3989 . . . . . . . . . . 11 ((π‘₯ βŠ† 𝐴 ∧ 𝐴 βŠ† On) β†’ π‘₯ βŠ† On)
43expcom 414 . . . . . . . . . 10 (𝐴 βŠ† On β†’ (π‘₯ βŠ† 𝐴 β†’ π‘₯ βŠ† On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 β†’ (π‘₯ βŠ† 𝐴 β†’ π‘₯ βŠ† On))
6 onsucuni 7812 . . . . . . . . 9 (π‘₯ βŠ† On β†’ π‘₯ βŠ† suc βˆͺ π‘₯)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 β†’ (π‘₯ βŠ† 𝐴 β†’ π‘₯ βŠ† suc βˆͺ π‘₯))
87adantrd 492 . . . . . . 7 (Lim 𝐴 β†’ ((π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ π‘₯ βŠ† suc βˆͺ π‘₯))
98ralimdv 3169 . . . . . 6 (Lim 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆ€π‘₯ ∈ 𝐡 π‘₯ βŠ† suc βˆͺ π‘₯))
10 uniiun 5060 . . . . . . 7 βˆͺ 𝐡 = βˆͺ π‘₯ ∈ 𝐡 π‘₯
11 ss2iun 5014 . . . . . . 7 (βˆ€π‘₯ ∈ 𝐡 π‘₯ βŠ† suc βˆͺ π‘₯ β†’ βˆͺ π‘₯ ∈ 𝐡 π‘₯ βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯)
1210, 11eqsstrid 4029 . . . . . 6 (βˆ€π‘₯ ∈ 𝐡 π‘₯ βŠ† suc βˆͺ π‘₯ β†’ βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯)
139, 12syl6 35 . . . . 5 (Lim 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯))
1413imp 407 . . . 4 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯)
15 cfslb.1 . . . . . . . . . 10 𝐴 ∈ V
1615cfslbn 10258 . . . . . . . . 9 ((Lim 𝐴 ∧ π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆͺ π‘₯ ∈ 𝐴)
17163expib 1122 . . . . . . . 8 (Lim 𝐴 β†’ ((π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆͺ π‘₯ ∈ 𝐴))
18 ordsucss 7802 . . . . . . . 8 (Ord 𝐴 β†’ (βˆͺ π‘₯ ∈ 𝐴 β†’ suc βˆͺ π‘₯ βŠ† 𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 β†’ ((π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ suc βˆͺ π‘₯ βŠ† 𝐴))
2019ralimdv 3169 . . . . . 6 (Lim 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴))
21 iunss 5047 . . . . . 6 (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴 ↔ βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴)
2220, 21syl6ibr 251 . . . . 5 (Lim 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴))
2322imp 407 . . . 4 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴)
24 sseq1 4006 . . . . . 6 (βˆͺ 𝐡 = 𝐴 β†’ (βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ↔ 𝐴 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯))
25 eqss 3996 . . . . . . 7 (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴 ↔ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯))
2625simplbi2com 503 . . . . . 6 (𝐴 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ β†’ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴 β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴))
2724, 26syl6bi 252 . . . . 5 (βˆͺ 𝐡 = 𝐴 β†’ (βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ β†’ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴 β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴)))
2827com3l 89 . . . 4 (βˆͺ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ β†’ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ βŠ† 𝐴 β†’ (βˆͺ 𝐡 = 𝐴 β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴)))
2914, 23, 28sylc 65 . . 3 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (βˆͺ 𝐡 = 𝐴 β†’ βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴))
30 limsuc 7834 . . . . . . . . 9 (Lim 𝐴 β†’ (βˆͺ π‘₯ ∈ 𝐴 ↔ suc βˆͺ π‘₯ ∈ 𝐴))
3117, 30sylibd 238 . . . . . . . 8 (Lim 𝐴 β†’ ((π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ suc βˆͺ π‘₯ ∈ 𝐴))
3231ralimdv 3169 . . . . . . 7 (Lim 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄)) β†’ βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ∈ 𝐴))
3332imp 407 . . . . . 6 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ∈ 𝐴)
34 r19.29 3114 . . . . . . . 8 ((βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ∈ 𝐴 ∧ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯) β†’ βˆƒπ‘₯ ∈ 𝐡 (suc βˆͺ π‘₯ ∈ 𝐴 ∧ 𝑦 = suc βˆͺ π‘₯))
35 eleq1 2821 . . . . . . . . . 10 (𝑦 = suc βˆͺ π‘₯ β†’ (𝑦 ∈ 𝐴 ↔ suc βˆͺ π‘₯ ∈ 𝐴))
3635biimparc 480 . . . . . . . . 9 ((suc βˆͺ π‘₯ ∈ 𝐴 ∧ 𝑦 = suc βˆͺ π‘₯) β†’ 𝑦 ∈ 𝐴)
3736rexlimivw 3151 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐡 (suc βˆͺ π‘₯ ∈ 𝐴 ∧ 𝑦 = suc βˆͺ π‘₯) β†’ 𝑦 ∈ 𝐴)
3834, 37syl 17 . . . . . . 7 ((βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ∈ 𝐴 ∧ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯) β†’ 𝑦 ∈ 𝐴)
3938ex 413 . . . . . 6 (βˆ€π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯ β†’ 𝑦 ∈ 𝐴))
4033, 39syl 17 . . . . 5 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯ β†’ 𝑦 ∈ 𝐴))
4140abssdv 4064 . . . 4 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} βŠ† 𝐴)
42 vuniex 7725 . . . . . . . 8 βˆͺ π‘₯ ∈ V
4342sucex 7790 . . . . . . 7 suc βˆͺ π‘₯ ∈ V
4443dfiun2 5035 . . . . . 6 βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯}
4544eqeq1i 2737 . . . . 5 (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴 ↔ βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} = 𝐴)
4615cfslb 10257 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} βŠ† 𝐴 ∧ βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} = 𝐴) β†’ (cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯})
47463expia 1121 . . . . 5 ((Lim 𝐴 ∧ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} βŠ† 𝐴) β†’ (βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} = 𝐴 β†’ (cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯}))
4845, 47biimtrid 241 . . . 4 ((Lim 𝐴 ∧ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} βŠ† 𝐴) β†’ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴 β†’ (cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯}))
4941, 48syldan 591 . . 3 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (βˆͺ π‘₯ ∈ 𝐡 suc βˆͺ π‘₯ = 𝐴 β†’ (cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯}))
50 eqid 2732 . . . . . . . . 9 (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) = (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯)
5150rnmpt 5952 . . . . . . . 8 ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯}
5243, 50fnmpti 6690 . . . . . . . . . 10 (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) Fn 𝐡
53 dffn4 6808 . . . . . . . . . 10 ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) Fn 𝐡 ↔ (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯))
5452, 53mpbi 229 . . . . . . . . 9 (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯)
55 relsdom 8942 . . . . . . . . . . 11 Rel β‰Ί
5655brrelex1i 5730 . . . . . . . . . 10 (𝐡 β‰Ί (cfβ€˜π΄) β†’ 𝐡 ∈ V)
57 breq1 5150 . . . . . . . . . . . 12 (𝑦 = 𝐡 β†’ (𝑦 β‰Ί (cfβ€˜π΄) ↔ 𝐡 β‰Ί (cfβ€˜π΄)))
58 foeq2 6799 . . . . . . . . . . . . 13 (𝑦 = 𝐡 β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝑦–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) ↔ (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯)))
59 breq2 5151 . . . . . . . . . . . . 13 (𝑦 = 𝐡 β†’ (ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝑦 ↔ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡))
6058, 59imbi12d 344 . . . . . . . . . . . 12 (𝑦 = 𝐡 β†’ (((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝑦–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝑦) ↔ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡)))
6157, 60imbi12d 344 . . . . . . . . . . 11 (𝑦 = 𝐡 β†’ ((𝑦 β‰Ί (cfβ€˜π΄) β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝑦–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝑦)) ↔ (𝐡 β‰Ί (cfβ€˜π΄) β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡))))
62 cfon 10246 . . . . . . . . . . . . 13 (cfβ€˜π΄) ∈ On
63 sdomdom 8972 . . . . . . . . . . . . 13 (𝑦 β‰Ί (cfβ€˜π΄) β†’ 𝑦 β‰Ό (cfβ€˜π΄))
64 ondomen 10028 . . . . . . . . . . . . 13 (((cfβ€˜π΄) ∈ On ∧ 𝑦 β‰Ό (cfβ€˜π΄)) β†’ 𝑦 ∈ dom card)
6562, 63, 64sylancr 587 . . . . . . . . . . . 12 (𝑦 β‰Ί (cfβ€˜π΄) β†’ 𝑦 ∈ dom card)
66 fodomnum 10048 . . . . . . . . . . . 12 (𝑦 ∈ dom card β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝑦–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝑦))
6765, 66syl 17 . . . . . . . . . . 11 (𝑦 β‰Ί (cfβ€˜π΄) β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝑦–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝑦))
6861, 67vtoclg 3556 . . . . . . . . . 10 (𝐡 ∈ V β†’ (𝐡 β‰Ί (cfβ€˜π΄) β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡)))
6956, 68mpcom 38 . . . . . . . . 9 (𝐡 β‰Ί (cfβ€˜π΄) β†’ ((π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯):𝐡–ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡))
7054, 69mpi 20 . . . . . . . 8 (𝐡 β‰Ί (cfβ€˜π΄) β†’ ran (π‘₯ ∈ 𝐡 ↦ suc βˆͺ π‘₯) β‰Ό 𝐡)
7151, 70eqbrtrrid 5183 . . . . . . 7 (𝐡 β‰Ί (cfβ€˜π΄) β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} β‰Ό 𝐡)
72 domtr 8999 . . . . . . 7 (((cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} ∧ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} β‰Ό 𝐡) β†’ (cfβ€˜π΄) β‰Ό 𝐡)
7371, 72sylan2 593 . . . . . 6 (((cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} ∧ 𝐡 β‰Ί (cfβ€˜π΄)) β†’ (cfβ€˜π΄) β‰Ό 𝐡)
74 domnsym 9095 . . . . . 6 ((cfβ€˜π΄) β‰Ό 𝐡 β†’ Β¬ 𝐡 β‰Ί (cfβ€˜π΄))
7573, 74syl 17 . . . . 5 (((cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} ∧ 𝐡 β‰Ί (cfβ€˜π΄)) β†’ Β¬ 𝐡 β‰Ί (cfβ€˜π΄))
7675pm2.01da 797 . . . 4 ((cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} β†’ Β¬ 𝐡 β‰Ί (cfβ€˜π΄))
7776a1i 11 . . 3 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ ((cfβ€˜π΄) β‰Ό {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐡 𝑦 = suc βˆͺ π‘₯} β†’ Β¬ 𝐡 β‰Ί (cfβ€˜π΄)))
7829, 49, 773syld 60 . 2 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (βˆͺ 𝐡 = 𝐴 β†’ Β¬ 𝐡 β‰Ί (cfβ€˜π΄)))
7978necon2ad 2955 1 ((Lim 𝐴 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰Ί (cfβ€˜π΄))) β†’ (𝐡 β‰Ί (cfβ€˜π΄) β†’ βˆͺ 𝐡 β‰  𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3947  βˆͺ cuni 4907  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363   Fn wfn 6535  β€“ontoβ†’wfo 6538  β€˜cfv 6540   β‰Ό cdom 8933   β‰Ί csdm 8934  cardccrd 9926  cfccf 9928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-card 9930  df-cf 9932  df-acn 9933
This theorem is referenced by:  tskuni  10774
  Copyright terms: Public domain W3C validator