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Mirrors > Home > MPE Home > Th. List > sspss | Structured version Visualization version GIF version |
Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.) |
Ref | Expression |
---|---|
sspss | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss2 4098 | . . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | 1 | simplbi2 500 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 → 𝐴 ⊊ 𝐵)) |
3 | 2 | con1d 145 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊊ 𝐵 → 𝐴 = 𝐵)) |
4 | 3 | orrd 863 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) |
5 | pssss 4108 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
6 | eqimss 4054 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | jaoi 857 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵) |
8 | 4, 7 | impbii 209 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∨ wo 847 = wceq 1537 ⊆ wss 3963 ⊊ wpss 3964 |
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-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-cleq 2727 df-ne 2939 df-ss 3980 df-pss 3983 |
This theorem is referenced by: sspsstri 4115 sspsstr 4118 psssstr 4119 ordsseleq 6415 sorpssuni 7751 sorpssint 7752 ssnnfi 9208 ackbij1b 10276 fin23lem40 10389 zorng 10542 psslinpr 11069 suplem2pr 11091 ressval3d 17292 ressval3dOLD 17293 mrissmrcd 17685 pgpssslw 19647 pgpfac1lem5 20114 idnghm 24780 slelss 27961 dfon2lem4 35768 finminlem 36301 lkrss2N 39151 dvh3dim3N 41432 ordsssucb 43325 |
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