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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 7728 | . 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 205 ∈ wcel 2099 Vcvv 3462 ⊊ wpss 3947 class class class wbr 5145 [⊊] crpss 7725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-xp 5680 df-rel 5681 df-rpss 7726 |
This theorem is referenced by: porpss 7730 sorpss 7731 fin23lem40 10385 compssiso 10408 isfin1-3 10420 fin12 10447 zorng 10538 fin2solem 37320 psshepw 43492 |
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