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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 4608 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | breq12 5143 | . . 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 3470 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 4 | canth2 9126 | . 2 ⊢ 𝑥 ≺ 𝒫 𝑥 |
6 | 3, 5 | vtoclg 3535 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 𝒫 cpw 4594 class class class wbr 5138 ≺ csdm 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-en 8936 df-dom 8937 df-sdom 8938 |
This theorem is referenced by: 2pwuninel 9128 2pwne 9129 pwfiOLD 9343 djulepw 10183 isfin32i 10356 fin34 10381 hsmexlem1 10417 canth3 10552 ondomon 10554 gchdomtri 10620 canthp1lem1 10643 canthp1lem2 10644 pwfseqlem5 10654 gchdjuidm 10659 gchxpidm 10660 gchpwdom 10661 gchaclem 10669 gchhar 10670 tsksdom 10747 |
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