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Theorem cfeq0 9191
 Description: Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfeq0 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cfeq0
Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 9182 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21eqeq1d 2726 . . 3 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅))
3 vex 3307 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2728 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
54anbi1d 743 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
65exbidv 1963 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
73, 6elab 3455 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
8 fveq2 6304 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9 cardidm 8898 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
108, 9syl6eq 2774 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11 eqeq2 2735 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
1210, 11mpbird 247 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1312adantr 472 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1413exlimiv 1971 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
157, 14sylbi 207 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
16 cardon 8883 . . . . . . 7 (card‘𝑣) ∈ On
1715, 16syl6eqelr 2812 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1817ssriv 3713 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
19 onint0 7113 . . . . 5 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))}))
2018, 19ax-mp 5 . . . 4 ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 0ex 4898 . . . . . 6 ∅ ∈ V
22 eqeq1 2728 . . . . . . . 8 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
2322anbi1d 743 . . . . . . 7 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2423exbidv 1963 . . . . . 6 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2521, 24elab 3455 . . . . 5 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
26 onss 7107 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
27 sstr 3717 . . . . . . . . . . . 12 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2827ancoms 468 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2926, 28sylan 489 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
30293adant2 1123 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦𝐴) → 𝑦 ⊆ On)
31303adant3r 1173 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
32 simp2 1129 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
33 simp3 1130 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
34 eqcom 2731 . . . . . . . . . . . 12 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35 vex 3307 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 8976 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
3735, 36mpan 708 . . . . . . . . . . . . 13 (𝑦 ⊆ On → 𝑦 ∈ dom card)
38 cardnueq0 8903 . . . . . . . . . . . . 13 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4034, 39syl5bb 272 . . . . . . . . . . 11 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4140biimpa 502 . . . . . . . . . 10 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42 sseq1 3732 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ⊆ 𝐴))
43 rexeq 3242 . . . . . . . . . . . . 13 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
4443ralbidv 3088 . . . . . . . . . . . 12 (𝑦 = ∅ → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4542, 44anbi12d 749 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)))
4645biimpa 502 . . . . . . . . . 10 ((𝑦 = ∅ ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4741, 46sylan 489 . . . . . . . . 9 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
48 rex0 4046 . . . . . . . . . . . . . 14 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4948rgenw 3026 . . . . . . . . . . . . 13 𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
50 r19.2z 4168 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan2 709 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
52 rexnal 3097 . . . . . . . . . . . 12 (∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤 ↔ ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5351, 52sylib 208 . . . . . . . . . . 11 (𝐴 ≠ ∅ → ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5453necon4ai 2927 . . . . . . . . . 10 (∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤𝐴 = ∅)
5554adantl 473 . . . . . . . . 9 ((∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤) → 𝐴 = ∅)
5647, 55syl 17 . . . . . . . 8 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
5731, 32, 33, 56syl21anc 1438 . . . . . . 7 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
58573expib 1116 . . . . . 6 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5958exlimdv 1974 . . . . 5 (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
6025, 59syl5bi 232 . . . 4 (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝐴 = ∅))
6120, 60syl5bi 232 . . 3 (𝐴 ∈ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ → 𝐴 = ∅))
622, 61sylbid 230 . 2 (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
63 fveq2 6304 . . 3 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
64 cf0 9186 . . 3 (cf‘∅) = ∅
6563, 64syl6eq 2774 . 2 (𝐴 = ∅ → (cf‘𝐴) = ∅)
6662, 65impbid1 215 1 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1596  ∃wex 1817   ∈ wcel 2103  {cab 2710   ≠ wne 2896  ∀wral 3014  ∃wrex 3015  Vcvv 3304   ⊆ wss 3680  ∅c0 4023  ∩ cint 4583  dom cdm 5218  Oncon0 5836  ‘cfv 6001  cardccrd 8874  cfccf 8876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-wrecs 7527  df-recs 7588  df-er 7862  df-en 8073  df-dom 8074  df-card 8878  df-cf 8880 This theorem is referenced by: (None)
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