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Mirrors > Home > MPE Home > Th. List > brrpss | Structured version Visualization version GIF version |
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
brrpss.a | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brrpss | ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrpss.a | . 2 ⊢ 𝐵 ∈ V | |
2 | brrpssg 7271 | . 2 ⊢ (𝐵 ∈ V → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2050 Vcvv 3415 ⊊ wpss 3832 class class class wbr 4930 [⊊] crpss 7268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 df-opab 4993 df-xp 5414 df-rel 5415 df-rpss 7269 |
This theorem is referenced by: porpss 7273 sorpss 7274 fin23lem40 9573 compssiso 9596 isfin1-3 9608 fin12 9635 zorng 9726 fin2solem 34319 psshepw 39497 |
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