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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 4514 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | breq12 5045 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) | |
3 | 1, 2 | mpdan 687 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) |
4 | vex 3404 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | canth2 8732 | . 2 ⊢ 𝑥 ≺ 𝒫 𝑥 |
6 | 3, 5 | vtoclg 3473 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 𝒫 cpw 4498 class class class wbr 5040 ≺ csdm 8566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-en 8568 df-dom 8569 df-sdom 8570 |
This theorem is referenced by: 2pwuninel 8734 2pwne 8735 pwfiOLD 8904 djulepw 9704 isfin32i 9877 fin34 9902 hsmexlem1 9938 canth3 10073 ondomon 10075 gchdomtri 10141 canthp1lem1 10164 canthp1lem2 10165 pwfseqlem5 10175 gchdjuidm 10180 gchxpidm 10181 gchpwdom 10182 gchaclem 10190 gchhar 10191 tsksdom 10268 |
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