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Mirrors > Home > MPE Home > Th. List > gch3 | Structured version Visualization version GIF version |
Description: An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gch3 | β’ (GCH = V β βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . 4 β’ ((GCH = V β§ π₯ β On) β π₯ β On) | |
2 | fvex 6859 | . . . . 5 β’ (β΅βπ₯) β V | |
3 | simpl 484 | . . . . 5 β’ ((GCH = V β§ π₯ β On) β GCH = V) | |
4 | 2, 3 | eleqtrrid 2841 | . . . 4 β’ ((GCH = V β§ π₯ β On) β (β΅βπ₯) β GCH) |
5 | fvex 6859 | . . . . 5 β’ (β΅βsuc π₯) β V | |
6 | 5, 3 | eleqtrrid 2841 | . . . 4 β’ ((GCH = V β§ π₯ β On) β (β΅βsuc π₯) β GCH) |
7 | gchaleph2 10616 | . . . 4 β’ ((π₯ β On β§ (β΅βπ₯) β GCH β§ (β΅βsuc π₯) β GCH) β (β΅βsuc π₯) β π« (β΅βπ₯)) | |
8 | 1, 4, 6, 7 | syl3anc 1372 | . . 3 β’ ((GCH = V β§ π₯ β On) β (β΅βsuc π₯) β π« (β΅βπ₯)) |
9 | 8 | ralrimiva 3140 | . 2 β’ (GCH = V β βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯)) |
10 | alephgch 10618 | . . . . . 6 β’ ((β΅βsuc π₯) β π« (β΅βπ₯) β (β΅βπ₯) β GCH) | |
11 | 10 | ralimi 3083 | . . . . 5 β’ (βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯) β βπ₯ β On (β΅βπ₯) β GCH) |
12 | alephfnon 10009 | . . . . . 6 β’ β΅ Fn On | |
13 | ffnfv 7070 | . . . . . 6 β’ (β΅:OnβΆGCH β (β΅ Fn On β§ βπ₯ β On (β΅βπ₯) β GCH)) | |
14 | 12, 13 | mpbiran 708 | . . . . 5 β’ (β΅:OnβΆGCH β βπ₯ β On (β΅βπ₯) β GCH) |
15 | 11, 14 | sylibr 233 | . . . 4 β’ (βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯) β β΅:OnβΆGCH) |
16 | 15 | frnd 6680 | . . 3 β’ (βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯) β ran β΅ β GCH) |
17 | gch2 10619 | . . 3 β’ (GCH = V β ran β΅ β GCH) | |
18 | 16, 17 | sylibr 233 | . 2 β’ (βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯) β GCH = V) |
19 | 9, 18 | impbii 208 | 1 β’ (GCH = V β βπ₯ β On (β΅βsuc π₯) β π« (β΅βπ₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 β wss 3914 π« cpw 4564 class class class wbr 5109 ran crn 5638 Oncon0 6321 suc csuc 6323 Fn wfn 6495 βΆwf 6496 βcfv 6500 β cen 8886 β΅cale 9880 GCHcgch 10564 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-seqom 8398 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-oexp 8422 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-har 9501 df-wdom 9509 df-cnf 9606 df-r1 9708 df-rank 9709 df-dju 9845 df-card 9883 df-aleph 9884 df-ac 10060 df-fin4 10231 df-gch 10565 |
This theorem is referenced by: (None) |
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