Description: In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd twice. This implies a special case of Theorem 4.2.2 in FaureFrolicher p. 87. (Contributed by David Moews, 1-May-2017)
Ref | Expression | ||
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Hypotheses | mreexfidimd.1 | |- ( ph -> A e. ( Moore ` X ) ) |
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mreexfidimd.2 | |- N = ( mrCls ` A ) |
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mreexfidimd.3 | |- I = ( mrInd ` A ) |
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mreexfidimd.4 | |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) ) |
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mreexfidimd.5 | |- ( ph -> S e. I ) |
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mreexfidimd.6 | |- ( ph -> T e. I ) |
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mreexfidimd.7 | |- ( ph -> S e. Fin ) |
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mreexfidimd.8 | |- ( ph -> ( N ` S ) = ( N ` T ) ) |
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Assertion | mreexfidimd | |- ( ph -> S ~~ T ) |
Step | Hyp | Ref | Expression |
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1 | mreexfidimd.1 | |- ( ph -> A e. ( Moore ` X ) ) |
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2 | mreexfidimd.2 | |- N = ( mrCls ` A ) |
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3 | mreexfidimd.3 | |- I = ( mrInd ` A ) |
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4 | mreexfidimd.4 | |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) ) |
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5 | mreexfidimd.5 | |- ( ph -> S e. I ) |
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6 | mreexfidimd.6 | |- ( ph -> T e. I ) |
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7 | mreexfidimd.7 | |- ( ph -> S e. Fin ) |
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8 | mreexfidimd.8 | |- ( ph -> ( N ` S ) = ( N ` T ) ) |
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9 | 3 1 5 | mrissd | |- ( ph -> S C_ X ) |
10 | 1 2 9 | mrcssidd | |- ( ph -> S C_ ( N ` S ) ) |
11 | 10 8 | sseqtrd | |- ( ph -> S C_ ( N ` T ) ) |
12 | 3 1 6 | mrissd | |- ( ph -> T C_ X ) |
13 | 7 | orcd | |- ( ph -> ( S e. Fin \/ T e. Fin ) ) |
14 | 1 2 3 4 11 12 13 5 | mreexdomd | |- ( ph -> S ~<_ T ) |
15 | 1 2 12 | mrcssidd | |- ( ph -> T C_ ( N ` T ) ) |
16 | 15 8 | sseqtrrd | |- ( ph -> T C_ ( N ` S ) ) |
17 | 7 | olcd | |- ( ph -> ( T e. Fin \/ S e. Fin ) ) |
18 | 1 2 3 4 16 9 17 6 | mreexdomd | |- ( ph -> T ~<_ S ) |
19 | sbth | |- ( ( S ~<_ T /\ T ~<_ S ) -> S ~~ T ) |
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20 | 14 18 19 | syl2anc | |- ( ph -> S ~~ T ) |