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Theorem cflim2 9037
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 3109 . . . . . . 7 (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴))
2 selpw 4142 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
3 limord 5748 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝐴 → Ord 𝐴)
4 ordsson 6943 . . . . . . . . . . . . . . . . . . . 20 (Ord 𝐴𝐴 ⊆ On)
5 sstr 3595 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
65expcom 451 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ On → (𝑦𝐴𝑦 ⊆ On))
73, 4, 63syl 18 . . . . . . . . . . . . . . . . . . 19 (Lim 𝐴 → (𝑦𝐴𝑦 ⊆ On))
87imp 445 . . . . . . . . . . . . . . . . . 18 ((Lim 𝐴𝑦𝐴) → 𝑦 ⊆ On)
983adant3 1079 . . . . . . . . . . . . . . . . 17 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ⊆ On)
10 ssel2 3582 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ On ∧ 𝑠𝑦) → 𝑠 ∈ On)
11 eloni 5697 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ On → Ord 𝑠)
12 ordirr 5705 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑠 → ¬ 𝑠𝑠)
1310, 11, 123syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑠𝑠)
14 ssel 3581 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑠 → (𝑠𝑦𝑠𝑠))
1514com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑠𝑦 → (𝑦𝑠𝑠𝑠))
1615adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (𝑦𝑠𝑠𝑠))
1713, 16mtod 189 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
189, 17sylan 488 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
19 simpl2 1063 . . . . . . . . . . . . . . . . 17 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦𝐴)
20 sstr 3595 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝐴𝑠) → 𝑦𝑠)
2119, 20sylan 488 . . . . . . . . . . . . . . . 16 ((((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) ∧ 𝐴𝑠) → 𝑦𝑠)
2218, 21mtand 690 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝐴𝑠)
23 simpl3 1064 . . . . . . . . . . . . . . . 16 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → 𝑦 = 𝐴)
2423sseq1d 3616 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ( 𝑦𝑠𝐴𝑠))
2522, 24mtbird 315 . . . . . . . . . . . . . 14 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
26 unissb 4440 . . . . . . . . . . . . . 14 ( 𝑦𝑠 ↔ ∀𝑡𝑦 𝑡𝑠)
2725, 26sylnib 318 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ ∀𝑡𝑦 𝑡𝑠)
2827nrexdv 2996 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 𝑡𝑠)
29 ssel 3581 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑠𝑦𝑠 ∈ On))
30 ssel 3581 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑡𝑦𝑡 ∈ On))
31 ontri1 5721 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
3231ancoms 469 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
33 vex 3192 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ V
34 vex 3192 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
3533, 34brcnv 5270 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 E 𝑠𝑠 E 𝑡)
36 epel 4993 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 E 𝑡𝑠𝑡)
3735, 36bitri 264 . . . . . . . . . . . . . . . . . . . 20 (𝑡 E 𝑠𝑠𝑡)
3837notbii 310 . . . . . . . . . . . . . . . . . . 19 𝑡 E 𝑠 ↔ ¬ 𝑠𝑡)
3932, 38syl6bbr 278 . . . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → ((𝑠𝑦𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠)))
4241impl 649 . . . . . . . . . . . . . . 15 (((𝑦 ⊆ On ∧ 𝑠𝑦) ∧ 𝑡𝑦) → (𝑡𝑠 ↔ ¬ 𝑡 E 𝑠))
4342ralbidva 2980 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (∀𝑡𝑦 𝑡𝑠 ↔ ∀𝑡𝑦 ¬ 𝑡 E 𝑠))
4443rexbidva 3043 . . . . . . . . . . . . 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 314 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
47 vex 3192 . . . . . . . . . . . . 13 𝑦 ∈ V
4847a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49 epweon 6937 . . . . . . . . . . . . . . . . . . 19 E We On
50 wess 5066 . . . . . . . . . . . . . . . . . . 19 (𝑦 ⊆ On → ( E We On → E We 𝑦))
5149, 50mpi 20 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → E We 𝑦)
52 weso 5070 . . . . . . . . . . . . . . . . . 18 ( E We 𝑦 → E Or 𝑦)
5351, 52syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → E Or 𝑦)
54 cnvso 5638 . . . . . . . . . . . . . . . . 17 ( E Or 𝑦 E Or 𝑦)
5553, 54sylib 208 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → E Or 𝑦)
5655adantr 481 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E Or 𝑦)
57 onssnum 8815 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
5847, 57mpan 705 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → 𝑦 ∈ dom card)
59 cardid2 8731 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
60 ensym 7957 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) ≈ 𝑦𝑦 ≈ (card‘𝑦))
6158, 59, 603syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
62 nnsdom 8503 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
63 ensdomtr 8048 . . . . . . . . . . . . . . . . 17 ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
6461, 62, 63syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
65 isfinite 8501 . . . . . . . . . . . . . . . 16 (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
6664, 65sylibr 224 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
67 wofi 8161 . . . . . . . . . . . . . . 15 (( E Or 𝑦𝑦 ∈ Fin) → E We 𝑦)
6856, 66, 67syl2anc 692 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
699, 68sylan 488 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E We 𝑦)
70 wefr 5069 . . . . . . . . . . . . 13 ( E We 𝑦 E Fr 𝑦)
7169, 70syl 17 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E Fr 𝑦)
72 ssid 3608 . . . . . . . . . . . . 13 𝑦𝑦
7372a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦𝑦)
74 unieq 4415 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
75 uni0 4436 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
7674, 75syl6eq 2671 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
77 eqeq1 2625 . . . . . . . . . . . . . . . . . 18 ( 𝑦 = 𝐴 → ( 𝑦 = ∅ ↔ 𝐴 = ∅))
7876, 77syl5ib 234 . . . . . . . . . . . . . . . . 17 ( 𝑦 = 𝐴 → (𝑦 = ∅ → 𝐴 = ∅))
79 nlim0 5747 . . . . . . . . . . . . . . . . . 18 ¬ Lim ∅
80 limeq 5699 . . . . . . . . . . . . . . . . . 18 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
8179, 80mtbiri 317 . . . . . . . . . . . . . . . . 17 (𝐴 = ∅ → ¬ Lim 𝐴)
8278, 81syl6 35 . . . . . . . . . . . . . . . 16 ( 𝑦 = 𝐴 → (𝑦 = ∅ → ¬ Lim 𝐴))
8382necon2ad 2805 . . . . . . . . . . . . . . 15 ( 𝑦 = 𝐴 → (Lim 𝐴𝑦 ≠ ∅))
8483impcom 446 . . . . . . . . . . . . . 14 ((Lim 𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
85843adant2 1078 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → 𝑦 ≠ ∅)
8685adantr 481 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ≠ ∅)
87 fri 5041 . . . . . . . . . . . 12 (((𝑦 ∈ V ∧ E Fr 𝑦) ∧ (𝑦𝑦𝑦 ≠ ∅)) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8848, 71, 73, 86, 87syl22anc 1324 . . . . . . . . . . 11 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ∃𝑠𝑦𝑡𝑦 ¬ 𝑡 E 𝑠)
8946, 88mtand 690 . . . . . . . . . 10 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ (card‘𝑦) ∈ ω)
90 cardon 8722 . . . . . . . . . . 11 (card‘𝑦) ∈ On
91 eloni 5697 . . . . . . . . . . 11 ((card‘𝑦) ∈ On → Ord (card‘𝑦))
92 ordom 7028 . . . . . . . . . . . 12 Ord ω
93 ordtri1 5720 . . . . . . . . . . . 12 ((Ord ω ∧ Ord (card‘𝑦)) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9492, 93mpan 705 . . . . . . . . . . 11 (Ord (card‘𝑦) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
9590, 91, 94mp2b 10 . . . . . . . . . 10 (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω)
9689, 95sylibr 224 . . . . . . . . 9 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
972, 96syl3an2b 1360 . . . . . . . 8 ((Lim 𝐴𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
98973expb 1263 . . . . . . 7 ((Lim 𝐴 ∧ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴)) → ω ⊆ (card‘𝑦))
991, 98sylan2b 492 . . . . . 6 ((Lim 𝐴𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}) → ω ⊆ (card‘𝑦))
10099ralrimiva 2961 . . . . 5 (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
101 ssiin 4541 . . . . 5 (ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) ↔ ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
102100, 101sylibr 224 . . . 4 (Lim 𝐴 → ω ⊆ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
103 cflim2.1 . . . . 5 𝐴 ∈ V
104103cflim3 9036 . . . 4 (Lim 𝐴 → (cf‘𝐴) = 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
105102, 104sseqtr4d 3626 . . 3 (Lim 𝐴 → ω ⊆ (cf‘𝐴))
106 fvex 6163 . . . . . . 7 (card‘𝑦) ∈ V
107106dfiin2 4526 . . . . . 6 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)}
108104, 107syl6eq 2671 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
109 cardlim 8750 . . . . . . . . 9 (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
110 sseq2 3611 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
111 limeq 5699 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (Lim 𝑥 ↔ Lim (card‘𝑦)))
112110, 111bibi12d 335 . . . . . . . . 9 (𝑥 = (card‘𝑦) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))))
113109, 112mpbiri 248 . . . . . . . 8 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
114113rexlimivw 3023 . . . . . . 7 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
115114ss2abi 3658 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)}
116 eleq1 2686 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
11790, 116mpbiri 248 . . . . . . . . 9 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
118117rexlimivw 3023 . . . . . . . 8 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
119118abssi 3661 . . . . . . 7 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
120 fvex 6163 . . . . . . . . 9 (cf‘𝐴) ∈ V
121108, 120syl6eqelr 2707 . . . . . . . 8 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
122 intex 4785 . . . . . . . 8 ({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
123121, 122sylibr 224 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅)
124 onint 6949 . . . . . . 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)})
125119, 123, 124sylancr 694 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
126115, 125sseldi 3585 . . . . 5 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
127108, 126eqeltrd 2698 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
128 sseq2 3611 . . . . . 6 (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
129 limeq 5699 . . . . . 6 (𝑥 = (cf‘𝐴) → (Lim 𝑥 ↔ Lim (cf‘𝐴)))
130128, 129bibi12d 335 . . . . 5 (𝑥 = (cf‘𝐴) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴))))
131120, 130elab 3337 . . . 4 ((cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)} ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
132127, 131sylib 208 . . 3 (Lim 𝐴 → (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
133105, 132mpbid 222 . 2 (Lim 𝐴 → Lim (cf‘𝐴))
134 eloni 5697 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
135 ordzsl 6999 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
136134, 135sylib 208 . . . . . 6 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
137 df-3or 1037 . . . . . . 7 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴))
138 orcom 402 . . . . . . 7 (((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴) ↔ (Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
139 df-or 385 . . . . . . 7 ((Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
140137, 138, 1393bitri 286 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
141136, 140sylib 208 . . . . 5 (𝐴 ∈ On → (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
142 fveq2 6153 . . . . . . . . 9 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
143 cf0 9025 . . . . . . . . 9 (cf‘∅) = ∅
144142, 143syl6eq 2671 . . . . . . . 8 (𝐴 = ∅ → (cf‘𝐴) = ∅)
145 limeq 5699 . . . . . . . 8 ((cf‘𝐴) = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
146144, 145syl 17 . . . . . . 7 (𝐴 = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
14779, 146mtbiri 317 . . . . . 6 (𝐴 = ∅ → ¬ Lim (cf‘𝐴))
148 1n0 7527 . . . . . . . . . 10 1𝑜 ≠ ∅
149 df1o2 7524 . . . . . . . . . . . 12 1𝑜 = {∅}
150149unieqi 4416 . . . . . . . . . . 11 1𝑜 = {∅}
151 0ex 4755 . . . . . . . . . . . 12 ∅ ∈ V
152151unisn 4422 . . . . . . . . . . 11 {∅} = ∅
153150, 152eqtri 2643 . . . . . . . . . 10 1𝑜 = ∅
154148, 153neeqtrri 2863 . . . . . . . . 9 1𝑜 1𝑜
155 limuni 5749 . . . . . . . . . 10 (Lim 1𝑜 → 1𝑜 = 1𝑜)
156155necon3ai 2815 . . . . . . . . 9 (1𝑜 1𝑜 → ¬ Lim 1𝑜)
157154, 156ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
158 fveq2 6153 . . . . . . . . . 10 (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
159 cfsuc 9031 . . . . . . . . . 10 (𝑥 ∈ On → (cf‘suc 𝑥) = 1𝑜)
160158, 159sylan9eqr 2677 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1𝑜)
161 limeq 5699 . . . . . . . . 9 ((cf‘𝐴) = 1𝑜 → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
162160, 161syl 17 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
163157, 162mtbiri 317 . . . . . . 7 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
164163rexlimiva 3022 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ Lim (cf‘𝐴))
165147, 164jaoi 394 . . . . 5 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
166141, 165syl6 35 . . . 4 (𝐴 ∈ On → (¬ Lim 𝐴 → ¬ Lim (cf‘𝐴)))
167166con4d 114 . . 3 (𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
168 cff 9022 . . . . . . . . 9 cf:On⟶On
169168fdmi 6014 . . . . . . . 8 dom cf = On
170169eleq2i 2690 . . . . . . 7 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
171 ndmfv 6180 . . . . . . 7 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
172170, 171sylnbir 321 . . . . . 6 𝐴 ∈ On → (cf‘𝐴) = ∅)
173172, 145syl 17 . . . . 5 𝐴 ∈ On → (Lim (cf‘𝐴) ↔ Lim ∅))
17479, 173mtbiri 317 . . . 4 𝐴 ∈ On → ¬ Lim (cf‘𝐴))
175174pm2.21d 118 . . 3 𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
176167, 175pm2.61i 176 . 2 (Lim (cf‘𝐴) → Lim 𝐴)
177133, 176impbii 199 1 (Lim 𝐴 ↔ Lim (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407   cint 4445   ciin 4491   class class class wbr 4618   E cep 4988   Or wor 4999   Fr wfr 5035   We wwe 5037  ccnv 5078  dom cdm 5079  Ord word 5686  Oncon0 5687  Lim wlim 5688  suc csuc 5689  cfv 5852  ωcom 7019  1𝑜c1o 7505  cen 7904  csdm 7906  Fincfn 7907  cardccrd 8713  cfccf 8715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cf 8719
This theorem is referenced by:  cfom  9038
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