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Theorem cardprc 9921
Description: The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 10502 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9485 to construct (effectively) (β„΅β€˜suc 𝐴) from (β„΅β€˜π΄), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
Assertion
Ref Expression
cardprc {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} βˆ‰ V

Proof of Theorem cardprc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . . 5 (π‘₯ = 𝑦 β†’ (cardβ€˜π‘₯) = (cardβ€˜π‘¦))
2 id 22 . . . . 5 (π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
31, 2eqeq12d 2749 . . . 4 (π‘₯ = 𝑦 β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜π‘¦) = 𝑦))
43cbvabv 2806 . . 3 {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} = {𝑦 ∣ (cardβ€˜π‘¦) = 𝑦}
54cardprclem 9920 . 2 Β¬ {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} ∈ V
65nelir 3049 1 {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} βˆ‰ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2710   βˆ‰ wnel 3046  Vcvv 3444  β€˜cfv 6497  cardccrd 9876
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-oi 9451  df-har 9498  df-card 9880
This theorem is referenced by:  alephprc  10040
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