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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 4111 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | breq12 4588 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) | |
3 | 1, 2 | mpdan 699 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥 ↔ 𝐴 ≺ 𝒫 𝐴)) |
4 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | canth2 7998 | . 2 ⊢ 𝑥 ≺ 𝒫 𝑥 |
6 | 3, 5 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 𝒫 cpw 4108 class class class wbr 4583 ≺ csdm 7840 |
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-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-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-ral 2901 df-rex 2902 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: 2pwuninel 8000 2pwne 8001 pwfi 8144 cdalepw 8901 isfin32i 9070 fin34 9095 hsmexlem1 9131 canth3 9262 ondomon 9264 gchdomtri 9330 canthp1lem1 9353 canthp1lem2 9354 pwfseqlem5 9364 gchcdaidm 9369 gchxpidm 9370 gchpwdom 9371 gchaclem 9379 gchhar 9380 tsksdom 9457 |
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