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Mirrors > Home > MPE Home > Th. List > dfpss3 | Structured version Visualization version GIF version |
Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
dfpss3 | ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss2 4098 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | eqss 4011 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | 2 | baib 535 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
4 | 3 | notbid 318 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴)) |
5 | 4 | pm5.32i 574 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
6 | 1, 5 | bitri 275 | 1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = 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-ex 1777 df-cleq 2727 df-ne 2939 df-ss 3980 df-pss 3983 |
This theorem is referenced by: pssirr 4113 pssn2lp 4114 ssnpss 4116 nsspssun 4274 pssdifcom1 4496 pssdifcom2 4497 php3 9247 php3OLD 9259 fincssdom 10361 reclem2pr 11086 ressval3d 17292 ressval3dOLD 17293 islbs3 21175 sltlpss 27960 chpsscon3 31532 chpssati 32392 fundmpss 35748 lpssat 38995 lssat 38998 dihglblem6 41323 pssnssi 45041 mbfpsssmf 46739 |
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