MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnum2 Structured version   Visualization version   GIF version

Theorem isnum2 9362
Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
isnum2 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem isnum2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardf2 9360 . . . 4 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
21fdmi 6517 . . 3 dom card = {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}
32eleq2i 2901 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦})
4 relen 8502 . . . . 5 Rel ≈
54brrelex2i 5602 . . . 4 (𝑥𝐴𝐴 ∈ V)
65rexlimivw 3279 . . 3 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
7 breq2 5061 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
87rexbidv 3294 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑥𝑦 ↔ ∃𝑥 ∈ On 𝑥𝐴))
96, 8elab3 3671 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦} ↔ ∃𝑥 ∈ On 𝑥𝐴)
103, 9bitri 276 1 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cen 8494  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-fun 6350  df-fn 6351  df-f 6352  df-en 8498  df-card 9356
This theorem is referenced by:  isnumi  9363  ennum  9364  xpnum  9368  cardval3  9369  dfac10c  9552  isfin7-2  9806  numth2  9881  inawinalem  10099
  Copyright terms: Public domain W3C validator