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Theorem cflim2 10286
Description: The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
Hypothesis
Ref Expression
cflim2.1 𝐴 ∈ V
Assertion
Ref Expression
cflim2 (Lim 𝐴 ↔ Lim (cf‘𝐴))

Proof of Theorem cflim2
Dummy variables 𝑠 𝑦 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabid 3449 . . . . . . 7 (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴))
2 velpw 4608 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
3 limord 6429 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝐴 → Ord 𝐴)
4 ordsson 7785 . . . . . . . . . . . . . . . . . . . 20 (Ord 𝐴𝐴 ⊆ On)
5 sstr 3988 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
65expcom 413 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑦𝐴𝑦 ⊆ On))
73, 4, 63syl 18 . . . . . . . . . . . . . . . . . . 19 (Lim 𝐴 → (𝑦𝐴𝑦 ⊆ On))
87imp 406 . . . . . . . . . . . . . . . . . 18 ((Lim 𝐴𝑦𝐴) → 𝑦 ⊆ On)
983adant3 1130 . . . . . . . . . . . . . . . . 17 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ⊆ On)
10 ssel2 3975 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ On ∧ 𝑠𝑦) → 𝑠 ∈ On)
11 eloni 6379 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ On → Ord 𝑠)
12 ordirr 6387 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑠 → ¬ 𝑠𝑠)
1310, 11, 123syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑠𝑠)
14 ssel 3973 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑠 → (𝑠𝑦𝑠𝑠))
1514com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑠𝑦 → (𝑦𝑠𝑠𝑠))
1615adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (𝑦𝑠𝑠𝑠))
1713, 16mtod 197 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
189, 17sylan 579 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
19 simpl2 1190 . . . . . . . . . . . . . . . . 17 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦𝐴)
20 sstr 3988 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝐴𝑠) → 𝑦𝑠)
2119, 20sylan 579 . . . . . . . . . . . . . . . 16 ((((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) ∧ 𝐴𝑠) → 𝑦𝑠)
2218, 21mtand 815 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝐴𝑠)
23 simpl3 1191 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦 = 𝐴)
2423sseq1d 4011 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ( 𝑦𝑠𝐴𝑠))
2522, 24mtbird 325 . . . . . . . . . . . . . 14 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
26 unissb 4942 . . . . . . . . . . . . . 14 ( 𝑦𝑠 ↔ ∀𝑡𝑦 𝑡𝑠)
2725, 26sylnib 328 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ ∀𝑡𝑦 𝑡𝑠)
2827nrexdv 3146 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 𝑡𝑠)
29 ssel 3973 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑠𝑦𝑠 ∈ On))
30 ssel 3973 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑡𝑦𝑡 ∈ On))
31 ontri1 6403 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
3231ancoms 458 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
33 vex 3475 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ V
34 vex 3475 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
3533, 34brcnv 5885 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 E 𝑠𝑠 E 𝑡)
36 epel 5585 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 E 𝑡𝑠𝑡)
3735, 36bitri 275 . . . . . . . . . . . . . . . . . . . 20 (𝑡 E 𝑠𝑠𝑡)
3837notbii 320 . . . . . . . . . . . . . . . . . . 19 𝑡 E 𝑠 ↔ ¬ 𝑠𝑡)
3932, 38bitr4di 289 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4039a1i 11 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4129, 30, 40syl2and 607 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → ((𝑠𝑦𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4241impl 455 . . . . . . . . . . . . . . 15 (((𝑦 ⊆ On ∧ 𝑠𝑦) ∧ 𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4342ralbidva 3172 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (∀𝑡𝑦 𝑡𝑠 ↔ ∀𝑡𝑦 ¬ 𝑡 E 𝑠))
4443rexbidva 3173 . . . . . . . . . . . . 13 (𝑦 ⊆ On → (∃𝑠𝑦𝑡𝑦 𝑡𝑠 ↔ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠))
459, 44syl 17 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → (∃𝑠𝑦𝑡𝑦 𝑡𝑠 ↔ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠))
4628, 45mtbid 324 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
47 vex 3475 . . . . . . . . . . . . 13 𝑦 ∈ V
4847a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49 epweon 7777 . . . . . . . . . . . . . . . . . 18 E We On
50 wess 5665 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → ( E We On → E We 𝑦))
5149, 50mpi 20 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → E We 𝑦)
52 weso 5669 . . . . . . . . . . . . . . . . 17 ( E We 𝑦 → E Or 𝑦)
5351, 52syl 17 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → E Or 𝑦)
54 cnvso 6292 . . . . . . . . . . . . . . . 16 ( E Or 𝑦 E Or 𝑦)
5553, 54sylib 217 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → E Or 𝑦)
56 onssnum 10063 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
5747, 56mpan 689 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → 𝑦 ∈ dom card)
58 cardid2 9976 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
59 ensym 9023 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) ≈ 𝑦𝑦 ≈ (card‘𝑦))
6057, 58, 593syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
61 nnsdom 9677 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
62 ensdomtr 9137 . . . . . . . . . . . . . . . . 17 ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
6360, 61, 62syl2an 595 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
64 isfinite 9675 . . . . . . . . . . . . . . . 16 (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
6563, 64sylibr 233 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
66 wofi 9316 . . . . . . . . . . . . . . 15 (( E Or 𝑦𝑦 ∈ Fin) → E We 𝑦)
6755, 65, 66syl2an2r 684 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
689, 67sylan 579 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
69 wefr 5668 . . . . . . . . . . . . 13 ( E We 𝑦 E Fr 𝑦)
7068, 69syl 17 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E Fr 𝑦)
71 ssidd 4003 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦𝑦)
72 unieq 4919 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
73 uni0 4938 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
7472, 73eqtrdi 2784 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
75 eqeq1 2732 . . . . . . . . . . . . . . . . . 18 ( 𝑦 = 𝐴 → ( 𝑦 = ∅ ↔ 𝐴 = ∅))
7674, 75imbitrid 243 . . . . . . . . . . . . . . . . 17 ( 𝑦 = 𝐴 → (𝑦 = ∅ → 𝐴 = ∅))
77 nlim0 6428 . . . . . . . . . . . . . . . . . 18 ¬ Lim ∅
78 limeq 6381 . . . . . . . . . . . . . . . . . 18 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
7977, 78mtbiri 327 . . . . . . . . . . . . . . . . 17 (𝐴 = ∅ → ¬ Lim 𝐴)
8076, 79syl6 35 . . . . . . . . . . . . . . . 16 ( 𝑦 = 𝐴 → (𝑦 = ∅ → ¬ Lim 𝐴))
8180necon2ad 2952 . . . . . . . . . . . . . . 15 ( 𝑦 = 𝐴 → (Lim 𝐴𝑦 ≠ ∅))
8281impcom 407 . . . . . . . . . . . . . 14 ((Lim 𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
83823adant2 1129 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
8483adantr 480 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ≠ ∅)
85 fri 5638 . . . . . . . . . . . 12 (((𝑦 ∈ V ∧ E Fr 𝑦) ∧ (𝑦𝑦𝑦 ≠ ∅)) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8648, 70, 71, 84, 85syl22anc 838 . . . . . . . . . . 11 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8746, 86mtand 815 . . . . . . . . . 10 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ (card‘𝑦) ∈ ω)
88 cardon 9967 . . . . . . . . . . 11 (card‘𝑦) ∈ On
89 eloni 6379 . . . . . . . . . . 11 ((card‘𝑦) ∈ On → Ord (card‘𝑦))
90 ordom 7880 . . . . . . . . . . . 12 Ord ω
91 ordtri1 6402 . . . . . . . . . . . 12 ((Ord ω ∧ Ord (card‘𝑦)) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9290, 91mpan 689 . . . . . . . . . . 11 (Ord (card‘𝑦) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9388, 89, 92mp2b 10 . . . . . . . . . 10 (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω)
9487, 93sylibr 233 . . . . . . . . 9 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
952, 94syl3an2b 1402 . . . . . . . 8 ((Lim 𝐴𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
96953expb 1118 . . . . . . 7 ((Lim 𝐴 ∧ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴)) → ω ⊆ (card‘𝑦))
971, 96sylan2b 593 . . . . . 6 ((Lim 𝐴𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}) → ω ⊆ (card‘𝑦))
9897ralrimiva 3143 . . . . 5 (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
99 ssiin 5058 . . . . 5 (ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) ↔ ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
10098, 99sylibr 233 . . . 4 (Lim 𝐴 → ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
101 cflim2.1 . . . . 5 𝐴 ∈ V
102101cflim3 10285 . . . 4 (Lim 𝐴 → (cf‘𝐴) = 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
103100, 102sseqtrrd 4021 . . 3 (Lim 𝐴 → ω ⊆ (cf‘𝐴))
104 fvex 6910 . . . . . . 7 (card‘𝑦) ∈ V
105104dfiin2 5037 . . . . . 6 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)}
106102, 105eqtrdi 2784 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
107 cardlim 9995 . . . . . . . . 9 (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
108 sseq2 4006 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
109 limeq 6381 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (Lim 𝑥 ↔ Lim (card‘𝑦)))
110108, 109bibi12d 345 . . . . . . . . 9 (𝑥 = (card‘𝑦) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))))
111107, 110mpbiri 258 . . . . . . . 8 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
112111rexlimivw 3148 . . . . . . 7 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
113112ss2abi 4061 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)}
114 eleq1 2817 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
11588, 114mpbiri 258 . . . . . . . . 9 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
116115rexlimivw 3148 . . . . . . . 8 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
117116abssi 4065 . . . . . . 7 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
118 fvex 6910 . . . . . . . . 9 (cf‘𝐴) ∈ V
119106, 118eqeltrrdi 2838 . . . . . . . 8 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
120 intex 5339 . . . . . . . 8 ({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
121119, 120sylibr 233 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅)
122 onint 7793 . . . . . . 7 (({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On ∧ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅) → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
123117, 121, 122sylancr 586 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
124113, 123sselid 3978 . . . . 5 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
125106, 124eqeltrd 2829 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
126 sseq2 4006 . . . . . 6 (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
127 limeq 6381 . . . . . 6 (𝑥 = (cf‘𝐴) → (Lim 𝑥 ↔ Lim (cf‘𝐴)))
128126, 127bibi12d 345 . . . . 5 (𝑥 = (cf‘𝐴) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴))))
129118, 128elab 3667 . . . 4 ((cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)} ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
130125, 129sylib 217 . . 3 (Lim 𝐴 → (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
131103, 130mpbid 231 . 2 (Lim 𝐴 → Lim (cf‘𝐴))
132 eloni 6379 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
133 ordzsl 7849 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
134132, 133sylib 217 . . . . . 6 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
135 df-3or 1086 . . . . . . 7 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴))
136 orcom 869 . . . . . . 7 (((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴) ↔ (Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
137 df-or 847 . . . . . . 7 ((Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
138135, 136, 1373bitri 297 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
139134, 138sylib 217 . . . . 5 (𝐴 ∈ On → (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
140 fveq2 6897 . . . . . . . . 9 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
141 cf0 10274 . . . . . . . . 9 (cf‘∅) = ∅
142140, 141eqtrdi 2784 . . . . . . . 8 (𝐴 = ∅ → (cf‘𝐴) = ∅)
143 limeq 6381 . . . . . . . 8 ((cf‘𝐴) = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
144142, 143syl 17 . . . . . . 7 (𝐴 = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
14577, 144mtbiri 327 . . . . . 6 (𝐴 = ∅ → ¬ Lim (cf‘𝐴))
146 1n0 8508 . . . . . . . . . 10 1o ≠ ∅
147 df1o2 8493 . . . . . . . . . . . 12 1o = {∅}
148147unieqi 4920 . . . . . . . . . . 11 1o = {∅}
149 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
150149unisn 4929 . . . . . . . . . . 11 {∅} = ∅
151148, 150eqtri 2756 . . . . . . . . . 10 1o = ∅
152146, 151neeqtrri 3011 . . . . . . . . 9 1o 1o
153 limuni 6430 . . . . . . . . . 10 (Lim 1o → 1o = 1o)
154153necon3ai 2962 . . . . . . . . 9 (1o 1o → ¬ Lim 1o)
155152, 154ax-mp 5 . . . . . . . 8 ¬ Lim 1o
156 fveq2 6897 . . . . . . . . . 10 (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
157 cfsuc 10280 . . . . . . . . . 10 (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
158156, 157sylan9eqr 2790 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1o)
159 limeq 6381 . . . . . . . . 9 ((cf‘𝐴) = 1o → (Lim (cf‘𝐴) ↔ Lim 1o))
160158, 159syl 17 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (Lim (cf‘𝐴) ↔ Lim 1o))
161155, 160mtbiri 327 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
162161rexlimiva 3144 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ Lim (cf‘𝐴))
163145, 162jaoi 856 . . . . 5 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
164139, 163syl6 35 . . . 4 (𝐴 ∈ On → (¬ Lim 𝐴 → ¬ Lim (cf‘𝐴)))
165164con4d 115 . . 3 (𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
166 cff 10271 . . . . . . . . 9 cf:On⟶On
167166fdmi 6734 . . . . . . . 8 dom cf = On
168167eleq2i 2821 . . . . . . 7 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
169 ndmfv 6932 . . . . . . 7 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
170168, 169sylnbir 331 . . . . . 6 𝐴 ∈ On → (cf‘𝐴) = ∅)
171170, 143syl 17 . . . . 5 𝐴 ∈ On → (Lim (cf‘𝐴) ↔ Lim ∅))
17277, 171mtbiri 327 . . . 4 𝐴 ∈ On → ¬ Lim (cf‘𝐴))
173172pm2.21d 121 . . 3 𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
174165, 173pm2.61i 182 . 2 (Lim (cf‘𝐴) → Lim 𝐴)
175131, 174impbii 208 1 (Lim 𝐴 ↔ Lim (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3o 1084  w3a 1085   = wceq 1534  wcel 2099  {cab 2705  wne 2937  wral 3058  wrex 3067  {crab 3429  Vcvv 3471  wss 3947  c0 4323  𝒫 cpw 4603  {csn 4629   cuni 4908   cint 4949   ciin 4997   class class class wbr 5148   E cep 5581   Or wor 5589   Fr wfr 5630   We wwe 5632  ccnv 5677  dom cdm 5678  Ord word 6368  Oncon0 6369  Lim wlim 6370  suc csuc 6371  cfv 6548  ωcom 7870  1oc1o 8479  cen 8960  csdm 8962  Fincfn 8963  cardccrd 9958  cfccf 9960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-card 9962  df-cf 9964
This theorem is referenced by:  cfom  10287
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