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Mirrors > Home > MPE Home > Th. List > canth2g | Structured version Visualization version GIF version |
Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.) |
Ref | Expression |
---|---|
canth2g | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4549 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | breq12 5079 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) | |
3 | 1, 2 | mpdan 684 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) |
4 | vex 3436 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | canth2 8917 | . 2 ⊢ 𝑥 ≺ 𝒫 𝑥 |
6 | 3, 5 | vtoclg 3505 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 𝒫 cpw 4533 class class class wbr 5074 ≺ csdm 8732 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-en 8734 df-dom 8735 df-sdom 8736 |
This theorem is referenced by: 2pwuninel 8919 2pwne 8920 pwfiOLD 9114 djulepw 9948 isfin32i 10121 fin34 10146 hsmexlem1 10182 canth3 10317 ondomon 10319 gchdomtri 10385 canthp1lem1 10408 canthp1lem2 10409 pwfseqlem5 10419 gchdjuidm 10424 gchxpidm 10425 gchpwdom 10426 gchaclem 10434 gchhar 10435 tsksdom 10512 |
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