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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 4673 | . . 3 β’ ((π΄ β© {π΅}) = β β Β¬ π΅ β π΄) | |
2 | cardon 9885 | . . . . . 6 β’ (cardβπ΄) β On | |
3 | 2 | onordi 6429 | . . . . 5 β’ Ord (cardβπ΄) |
4 | orddisj 6356 | . . . . 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 10488 | . . . . . 6 β’ (cardβπ΄) β π΄ |
8 | 7 | ensymi 8947 | . . . . 5 β’ π΄ β (cardβπ΄) |
9 | unsnen.2 | . . . . . 6 β’ π΅ β V | |
10 | fvex 6856 | . . . . . 6 β’ (cardβπ΄) β V | |
11 | en2sn 8988 | . . . . . 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 8993 | . . . . 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 6324 | . 2 β’ suc (cardβπ΄) = ((cardβπ΄) βͺ {(cardβπ΄)}) | |
18 | 16, 17 | breqtrrdi 5148 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 βͺ cun 3909 β© cin 3910 β c0 4283 {csn 4587 class class class wbr 5106 Ord word 6317 suc csuc 6320 βcfv 6497 β cen 8883 cardccrd 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-ac2 10404 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-er 8651 df-en 8887 df-card 9880 df-ac 10057 |
This theorem is referenced by: (None) |
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