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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 7744 | . 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 2106 Vcvv 3478 ⊊ wpss 3964 class class class wbr 5148 [⊊] crpss 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-rpss 7742 |
This theorem is referenced by: porpss 7746 sorpss 7747 fin23lem40 10389 compssiso 10412 isfin1-3 10424 fin12 10451 zorng 10542 fin2solem 37593 psshepw 43778 |
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