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Mirrors > Home > MPE Home > Th. List > unsnen | Structured version Visualization version GIF version |
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
unsnen.1 | β’ π΄ β V |
unsnen.2 | β’ π΅ β V |
Ref | Expression |
---|---|
unsnen | β’ (Β¬ π΅ β π΄ β (π΄ βͺ {π΅}) β suc (cardβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn 4710 | . . 3 β’ ((π΄ β© {π΅}) = β β Β¬ π΅ β π΄) | |
2 | cardon 9938 | . . . . . 6 β’ (cardβπ΄) β On | |
3 | 2 | onordi 6468 | . . . . 5 β’ Ord (cardβπ΄) |
4 | orddisj 6395 | . . . . 5 β’ (Ord (cardβπ΄) β ((cardβπ΄) β© {(cardβπ΄)}) = β ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ ((cardβπ΄) β© {(cardβπ΄)}) = β |
6 | unsnen.1 | . . . . . . 7 β’ π΄ β V | |
7 | 6 | cardid 10541 | . . . . . 6 β’ (cardβπ΄) β π΄ |
8 | 7 | ensymi 8999 | . . . . 5 β’ π΄ β (cardβπ΄) |
9 | unsnen.2 | . . . . . 6 β’ π΅ β V | |
10 | fvex 6897 | . . . . . 6 β’ (cardβπ΄) β V | |
11 | en2sn 9040 | . . . . . 6 β’ ((π΅ β V β§ (cardβπ΄) β V) β {π΅} β {(cardβπ΄)}) | |
12 | 9, 10, 11 | mp2an 689 | . . . . 5 β’ {π΅} β {(cardβπ΄)} |
13 | unen 9045 | . . . . 5 β’ (((π΄ β (cardβπ΄) β§ {π΅} β {(cardβπ΄)}) β§ ((π΄ β© {π΅}) = β β§ ((cardβπ΄) β© {(cardβπ΄)}) = β )) β (π΄ βͺ {π΅}) β ((cardβπ΄) βͺ {(cardβπ΄)})) | |
14 | 8, 12, 13 | mpanl12 699 | . . . 4 β’ (((π΄ β© {π΅}) = β β§ ((cardβπ΄) β© {(cardβπ΄)}) = β ) β (π΄ βͺ {π΅}) β ((cardβπ΄) βͺ {(cardβπ΄)})) |
15 | 5, 14 | mpan2 688 | . . 3 β’ ((π΄ β© {π΅}) = β β (π΄ βͺ {π΅}) β ((cardβπ΄) βͺ {(cardβπ΄)})) |
16 | 1, 15 | sylbir 234 | . 2 β’ (Β¬ π΅ β π΄ β (π΄ βͺ {π΅}) β ((cardβπ΄) βͺ {(cardβπ΄)})) |
17 | df-suc 6363 | . 2 β’ suc (cardβπ΄) = ((cardβπ΄) βͺ {(cardβπ΄)}) | |
18 | 16, 17 | breqtrrdi 5183 | 1 β’ (Β¬ π΅ β π΄ β (π΄ βͺ {π΅}) β suc (cardβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 βͺ cun 3941 β© cin 3942 β c0 4317 {csn 4623 class class class wbr 5141 Ord word 6356 suc csuc 6359 βcfv 6536 β cen 8935 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-ac2 10457 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-er 8702 df-en 8939 df-card 9933 df-ac 10110 |
This theorem is referenced by: (None) |
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