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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 4383 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | breq12 4880 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) | |
3 | 1, 2 | mpdan 678 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) |
4 | vex 3417 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | canth2 8388 | . 2 ⊢ 𝑥 ≺ 𝒫 𝑥 |
6 | 3, 5 | vtoclg 3482 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 𝒫 cpw 4380 class class class wbr 4875 ≺ csdm 8227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-en 8229 df-dom 8230 df-sdom 8231 |
This theorem is referenced by: 2pwuninel 8390 2pwne 8391 pwfi 8536 cdalepw 9340 isfin32i 9509 fin34 9534 hsmexlem1 9570 canth3 9705 ondomon 9707 gchdomtri 9773 canthp1lem1 9796 canthp1lem2 9797 pwfseqlem5 9807 gchcdaidm 9812 gchxpidm 9813 gchpwdom 9814 gchaclem 9822 gchhar 9823 tsksdom 9900 |
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