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Mirrors > Home > MPE Home > Th. List > cardprc | Structured version Visualization version GIF version |
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 9262 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 8332 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.) |
Ref | Expression |
---|---|
cardprc | ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦)) | |
2 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
3 | 1, 2 | eqeq12d 2625 | . . . 4 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦)) |
4 | 3 | cbvabv 2734 | . . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦} |
5 | 4 | cardprclem 8688 | . 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V |
6 | 5 | nelir 2886 | 1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {cab 2596 ∉ wnel 2781 Vcvv 3173 ‘cfv 5804 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-oi 8298 df-har 8346 df-card 8648 |
This theorem is referenced by: alephprc 8805 |
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