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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 4716 | . . 3 β’ ((π΄ β© {π΅}) = β β Β¬ π΅ β π΄) | |
2 | cardon 9968 | . . . . . 6 β’ (cardβπ΄) β On | |
3 | 2 | onordi 6480 | . . . . 5 β’ Ord (cardβπ΄) |
4 | orddisj 6407 | . . . . 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 10571 | . . . . . 6 β’ (cardβπ΄) β π΄ |
8 | 7 | ensymi 9025 | . . . . 5 β’ π΄ β (cardβπ΄) |
9 | unsnen.2 | . . . . . 6 β’ π΅ β V | |
10 | fvex 6910 | . . . . . 6 β’ (cardβπ΄) β V | |
11 | en2sn 9066 | . . . . . 6 β’ ((π΅ β V β§ (cardβπ΄) β V) β {π΅} β {(cardβπ΄)}) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 β’ {π΅} β {(cardβπ΄)} |
13 | unen 9071 | . . . . 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 701 | . . . 4 β’ (((π΄ β© {π΅}) = β β§ ((cardβπ΄) β© {(cardβπ΄)}) = β ) β (π΄ βͺ {π΅}) β ((cardβπ΄) βͺ {(cardβπ΄)})) |
15 | 5, 14 | mpan2 690 | . . 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 6375 | . 2 β’ suc (cardβπ΄) = ((cardβπ΄) βͺ {(cardβπ΄)}) | |
18 | 16, 17 | breqtrrdi 5190 | 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 1534 β wcel 2099 Vcvv 3471 βͺ cun 3945 β© cin 3946 β c0 4323 {csn 4629 class class class wbr 5148 Ord word 6368 suc csuc 6371 βcfv 6548 β cen 8961 cardccrd 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-ac2 10487 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-er 8725 df-en 8965 df-card 9963 df-ac 10140 |
This theorem is referenced by: (None) |
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