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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 4061 | . . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | 1 | simplbi2 503 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 → 𝐴 ⊊ 𝐵)) |
3 | 2 | con1d 147 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 ⊊ 𝐵 → 𝐴 = 𝐵)) |
4 | 3 | orrd 859 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) |
5 | pssss 4071 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
6 | eqimss 4022 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | jaoi 853 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵) |
8 | 4, 7 | impbii 211 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∨ wo 843 = wceq 1533 ⊆ wss 3935 ⊊ wpss 3936 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-in 3942 df-ss 3951 df-pss 3953 |
This theorem is referenced by: sspsstri 4078 sspsstr 4081 psssstr 4082 ordsseleq 6214 sorpssuni 7452 sorpssint 7453 ssnnfi 8731 ackbij1b 9655 fin23lem40 9767 zorng 9920 psslinpr 10447 suplem2pr 10469 ressval3d 16555 mrissmrcd 16905 pgpssslw 18733 pgpfac1lem5 19195 idnghm 23346 dfon2lem4 33026 finminlem 33661 lkrss2N 36299 dvh3dim3N 38579 |
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