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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 7453 | . 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 208 ∈ wcel 2114 Vcvv 3496 ⊊ wpss 3939 class class class wbr 5068 [⊊] crpss 7450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-rpss 7451 |
This theorem is referenced by: porpss 7455 sorpss 7456 fin23lem40 9775 compssiso 9798 isfin1-3 9810 fin12 9837 zorng 9928 fin2solem 34880 psshepw 40141 |
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