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Theorem cflim2 9045
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 3110 . . . . . . 7 (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴))
2 selpw 4143 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
3 limord 5753 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝐴 → Ord 𝐴)
4 ordsson 6951 . . . . . . . . . . . . . . . . . . . 20 (Ord 𝐴𝐴 ⊆ On)
5 sstr 3596 . . . . . . . . . . . . . . . . . . . . 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 3583 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ On ∧ 𝑠𝑦) → 𝑠 ∈ On)
11 eloni 5702 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ On → Ord 𝑠)
12 ordirr 5710 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑠 → ¬ 𝑠𝑠)
1310, 11, 123syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ On ∧ 𝑠𝑦) → ¬ 𝑠𝑠)
14 ssel 3582 . . . . . . . . . . . . . . . . . . . 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 3596 . . . . . . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . 15 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ( 𝑦𝑠𝐴𝑠))
2522, 24mtbird 315 . . . . . . . . . . . . . 14 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ 𝑦𝑠)
26 unissb 4442 . . . . . . . . . . . . . 14 ( 𝑦𝑠 ↔ ∀𝑡𝑦 𝑡𝑠)
2725, 26sylnib 318 . . . . . . . . . . . . 13 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ 𝑠𝑦) → ¬ ∀𝑡𝑦 𝑡𝑠)
2827nrexdv 2997 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) → ¬ ∃𝑠𝑦𝑡𝑦 𝑡𝑠)
29 ssel 3582 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑠𝑦𝑠 ∈ On))
30 ssel 3582 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → (𝑡𝑦𝑡 ∈ On))
31 ontri1 5726 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
3231ancoms 469 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡𝑠 ↔ ¬ 𝑠𝑡))
33 vex 3193 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ V
34 vex 3193 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
3533, 34brcnv 5275 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 E 𝑠𝑠 E 𝑡)
36 epel 4998 . . . . . . . . . . . . . . . . . . . . 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 2981 . . . . . . . . . . . . . 14 ((𝑦 ⊆ On ∧ 𝑠𝑦) → (∀𝑡𝑦 𝑡𝑠 ↔ ∀𝑡𝑦 ¬ 𝑡 E 𝑠))
4443rexbidva 3044 . . . . . . . . . . . . 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 3193 . . . . . . . . . . . . 13 𝑦 ∈ V
4847a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49 epweon 6945 . . . . . . . . . . . . . . . . . . 19 E We On
50 wess 5071 . . . . . . . . . . . . . . . . . . 19 (𝑦 ⊆ On → ( E We On → E We 𝑦))
5149, 50mpi 20 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → E We 𝑦)
52 weso 5075 . . . . . . . . . . . . . . . . . 18 ( E We 𝑦 → E Or 𝑦)
5351, 52syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → E Or 𝑦)
54 cnvso 5643 . . . . . . . . . . . . . . . . 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 8823 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
5847, 57mpan 705 . . . . . . . . . . . . . . . . . 18 (𝑦 ⊆ On → 𝑦 ∈ dom card)
59 cardid2 8739 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
60 ensym 7965 . . . . . . . . . . . . . . . . . 18 ((card‘𝑦) ≈ 𝑦𝑦 ≈ (card‘𝑦))
6158, 59, 603syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
62 nnsdom 8511 . . . . . . . . . . . . . . . . 17 ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
63 ensdomtr 8056 . . . . . . . . . . . . . . . . 17 ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
6461, 62, 63syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
65 isfinite 8509 . . . . . . . . . . . . . . . 16 (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
6664, 65sylibr 224 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
67 wofi 8169 . . . . . . . . . . . . . . 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 5074 . . . . . . . . . . . . 13 ( E We 𝑦 E Fr 𝑦)
7169, 70syl 17 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → E Fr 𝑦)
72 ssid 3609 . . . . . . . . . . . . 13 𝑦𝑦
7372a1i 11 . . . . . . . . . . . 12 (((Lim 𝐴𝑦𝐴 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦𝑦)
74 unieq 4417 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
75 uni0 4438 . . . . . . . . . . . . . . . . . . 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 5752 . . . . . . . . . . . . . . . . . 18 ¬ Lim ∅
80 limeq 5704 . . . . . . . . . . . . . . . . . 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 5046 . . . . . . . . . . . 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 8730 . . . . . . . . . . 11 (card‘𝑦) ∈ On
91 eloni 5702 . . . . . . . . . . 11 ((card‘𝑦) ∈ On → Ord (card‘𝑦))
92 ordom 7036 . . . . . . . . . . . 12 Ord ω
93 ordtri1 5725 . . . . . . . . . . . 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 2962 . . . . 5 (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
101 ssiin 4543 . . . . 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 9044 . . . 4 (Lim 𝐴 → (cf‘𝐴) = 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴} (card‘𝑦))
105102, 104sseqtr4d 3627 . . 3 (Lim 𝐴 → ω ⊆ (cf‘𝐴))
106 fvex 6168 . . . . . . 7 (card‘𝑦) ∈ V
107106dfiin2 4528 . . . . . 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 8758 . . . . . . . . 9 (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
110 sseq2 3612 . . . . . . . . . 10 (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
111 limeq 5704 . . . . . . . . . 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 3024 . . . . . . 7 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
115114ss2abi 3659 . . . . . 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 3024 . . . . . . . 8 (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
119118abssi 3662 . . . . . . 7 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
120 fvex 6168 . . . . . . . . 9 (cf‘𝐴) ∈ V
121108, 120syl6eqelr 2707 . . . . . . . 8 (Lim 𝐴 {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
122 intex 4790 . . . . . . . 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 6957 . . . . . . 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 3586 . . . . 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 3612 . . . . . 6 (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
129 limeq 5704 . . . . . 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 3338 . . . 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 5702 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
135 ordzsl 7007 . . . . . . 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 6158 . . . . . . . . 9 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
143 cf0 9033 . . . . . . . . 9 (cf‘∅) = ∅
144142, 143syl6eq 2671 . . . . . . . 8 (𝐴 = ∅ → (cf‘𝐴) = ∅)
145 limeq 5704 . . . . . . . 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 7535 . . . . . . . . . 10 1𝑜 ≠ ∅
149 df1o2 7532 . . . . . . . . . . . 12 1𝑜 = {∅}
150149unieqi 4418 . . . . . . . . . . 11 1𝑜 = {∅}
151 0ex 4760 . . . . . . . . . . . 12 ∅ ∈ V
152151unisn 4424 . . . . . . . . . . 11 {∅} = ∅
153150, 152eqtri 2643 . . . . . . . . . 10 1𝑜 = ∅
154148, 153neeqtrri 2863 . . . . . . . . 9 1𝑜 1𝑜
155 limuni 5754 . . . . . . . . . 10 (Lim 1𝑜 → 1𝑜 = 1𝑜)
156155necon3ai 2815 . . . . . . . . 9 (1𝑜 1𝑜 → ¬ Lim 1𝑜)
157154, 156ax-mp 5 . . . . . . . 8 ¬ Lim 1𝑜
158 fveq2 6158 . . . . . . . . . 10 (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
159 cfsuc 9039 . . . . . . . . . 10 (𝑥 ∈ On → (cf‘suc 𝑥) = 1𝑜)
160158, 159sylan9eqr 2677 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1𝑜)
161 limeq 5704 . . . . . . . . 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 3023 . . . . . 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 9030 . . . . . . . . 9 cf:On⟶On
169168fdmi 6019 . . . . . . . 8 dom cf = On
170169eleq2i 2690 . . . . . . 7 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
171 ndmfv 6185 . . . . . . 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 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897  𝒫 cpw 4136  {csn 4155   cuni 4409   cint 4447   ciin 4493   class class class wbr 4623   E cep 4993   Or wor 5004   Fr wfr 5040   We wwe 5042  ccnv 5083  dom cdm 5084  Ord word 5691  Oncon0 5692  Lim wlim 5693  suc csuc 5694  cfv 5857  ωcom 7027  1𝑜c1o 7513  cen 7912  csdm 7914  Fincfn 7915  cardccrd 8721  cfccf 8723
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cf 8727
This theorem is referenced by:  cfom  9046
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