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Theorem cfeq0 9667
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 9658 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21eqeq1d 2828 . . 3 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅))
3 vex 3503 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2830 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
54anbi1d 629 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
65exbidv 1915 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
73, 6elab 3671 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
8 fveq2 6667 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9 cardidm 9377 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
108, 9syl6eq 2877 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11 eqeq2 2838 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
1210, 11mpbird 258 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1312adantr 481 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1413exlimiv 1924 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
157, 14sylbi 218 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
16 cardon 9362 . . . . . . 7 (card‘𝑣) ∈ On
1715, 16syl6eqelr 2927 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1817ssriv 3975 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
19 onint0 7499 . . . . 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 5208 . . . . . 6 ∅ ∈ V
22 eqeq1 2830 . . . . . . . 8 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
2322anbi1d 629 . . . . . . 7 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2423exbidv 1915 . . . . . 6 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2521, 24elab 3671 . . . . 5 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
26 onss 7493 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
27 sstr 3979 . . . . . . . . . . . 12 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2827ancoms 459 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2926, 28sylan 580 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
30293adant2 1125 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦𝐴) → 𝑦 ⊆ On)
31303adant3r 1175 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
32 simp2 1131 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
33 simp3 1132 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
34 eqcom 2833 . . . . . . . . . . . 12 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35 vex 3503 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 9455 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
3735, 36mpan 686 . . . . . . . . . . . . 13 (𝑦 ⊆ On → 𝑦 ∈ dom card)
38 cardnueq0 9382 . . . . . . . . . . . . 13 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4034, 39syl5bb 284 . . . . . . . . . . 11 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4140biimpa 477 . . . . . . . . . 10 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42 sseq1 3996 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ⊆ 𝐴))
43 rexeq 3412 . . . . . . . . . . . . 13 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
4443ralbidv 3202 . . . . . . . . . . . 12 (𝑦 = ∅ → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4542, 44anbi12d 630 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)))
4645biimpa 477 . . . . . . . . . 10 ((𝑦 = ∅ ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4741, 46sylan 580 . . . . . . . . 9 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
48 rex0 4321 . . . . . . . . . . . . 13 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4948rgenw 3155 . . . . . . . . . . . 12 𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
50 r19.2z 4443 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan2 687 . . . . . . . . . . 11 (𝐴 ≠ ∅ → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
52 rexnal 3243 . . . . . . . . . . 11 (∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤 ↔ ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5351, 52sylib 219 . . . . . . . . . 10 (𝐴 ≠ ∅ → ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5453necon4ai 3052 . . . . . . . . 9 (∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤𝐴 = ∅)
5547, 54simpl2im 504 . . . . . . . 8 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
5631, 32, 33, 55syl21anc 835 . . . . . . 7 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
57563expib 1116 . . . . . 6 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5857exlimdv 1927 . . . . 5 (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5925, 58syl5bi 243 . . . 4 (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝐴 = ∅))
6020, 59syl5bi 243 . . 3 (𝐴 ∈ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ → 𝐴 = ∅))
612, 60sylbid 241 . 2 (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62 fveq2 6667 . . 3 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63 cf0 9662 . . 3 (cf‘∅) = ∅
6462, 63syl6eq 2877 . 2 (𝐴 = ∅ → (cf‘𝐴) = ∅)
6561, 64impbid1 226 1 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wne 3021  wral 3143  wrex 3144  Vcvv 3500  wss 3940  c0 4295   cint 4874  dom cdm 5554  Oncon0 6189  cfv 6352  cardccrd 9353  cfccf 9355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-wrecs 7938  df-recs 7999  df-er 8279  df-en 8499  df-dom 8500  df-card 9357  df-cf 9359
This theorem is referenced by: (None)
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