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Theorem isnum2 9937
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 ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
Distinct variable group:   π‘₯,𝐴

Proof of Theorem isnum2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardf2 9935 . . . 4 card:{𝑦 ∣ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝑦}⟢On
21fdmi 6727 . . 3 dom card = {𝑦 ∣ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝑦}
32eleq2i 2826 . 2 (𝐴 ∈ dom card ↔ 𝐴 ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝑦})
4 relen 8941 . . . . 5 Rel β‰ˆ
54brrelex2i 5732 . . . 4 (π‘₯ β‰ˆ 𝐴 β†’ 𝐴 ∈ V)
65rexlimivw 3152 . . 3 (βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴 β†’ 𝐴 ∈ V)
7 breq2 5152 . . . 4 (𝑦 = 𝐴 β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ β‰ˆ 𝐴))
87rexbidv 3179 . . 3 (𝑦 = 𝐴 β†’ (βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝑦 ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴))
96, 8elab3 3676 . 2 (𝐴 ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝑦} ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
103, 9bitri 275 1 (𝐴 ∈ dom card ↔ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  Vcvv 3475   class class class wbr 5148  dom cdm 5676  Oncon0 6362   β‰ˆ cen 8933  cardccrd 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-fun 6543  df-fn 6544  df-f 6545  df-en 8937  df-card 9931
This theorem is referenced by:  isnumi  9938  ennum  9939  xpnum  9943  cardval3  9944  dfac10c  10130  isfin7-2  10388  numth2  10463  inawinalem  10681
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