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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 7746 | . 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 206 ∈ wcel 2107 Vcvv 3479 ⊊ wpss 3951 class class class wbr 5142 [⊊] crpss 7743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-rpss 7744 | 
| This theorem is referenced by: porpss 7748 sorpss 7749 fin23lem40 10392 compssiso 10415 isfin1-3 10427 fin12 10454 zorng 10545 fin2solem 37614 psshepw 43806 | 
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