![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > prv0 | Structured version Visualization version GIF version |
Description: Every wff encoded as π is true in an "empty model" (π = β ). Since β§ is defined in terms of the interpretations making the given formula true, it is not defined on the "empty model", since there are no interpretations. In particular, the empty set on the LHS of β§ should not be interpreted as the empty model, because βπ₯π₯ = π₯ is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.) |
Ref | Expression |
---|---|
prv0 | β’ (π β (FmlaβΟ) β β β§π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sate0 34475 | . . 3 β’ (π β (FmlaβΟ) β (β Satβ π) = (((β Sat β )βΟ)βπ)) | |
2 | peano1 7881 | . . . . . . . . . 10 β’ β β Ο | |
3 | 2 | n0ii 4336 | . . . . . . . . 9 β’ Β¬ Ο = β |
4 | 3 | intnan 487 | . . . . . . . 8 β’ Β¬ (π₯ = β β§ Ο = β ) |
5 | 4 | a1i 11 | . . . . . . 7 β’ (π β (FmlaβΟ) β Β¬ (π₯ = β β§ Ο = β )) |
6 | f00 6773 | . . . . . . 7 β’ (π₯:ΟβΆβ β (π₯ = β β§ Ο = β )) | |
7 | 5, 6 | sylnibr 328 | . . . . . 6 β’ (π β (FmlaβΟ) β Β¬ π₯:ΟβΆβ ) |
8 | 0ex 5307 | . . . . . . . 8 β’ β β V | |
9 | 8, 8 | pm3.2i 471 | . . . . . . 7 β’ (β β V β§ β β V) |
10 | satfvel 34472 | . . . . . . 7 β’ (((β β V β§ β β V) β§ π β (FmlaβΟ) β§ π₯ β (((β Sat β )βΟ)βπ)) β π₯:ΟβΆβ ) | |
11 | 9, 10 | mp3an1 1448 | . . . . . 6 β’ ((π β (FmlaβΟ) β§ π₯ β (((β Sat β )βΟ)βπ)) β π₯:ΟβΆβ ) |
12 | 7, 11 | mtand 814 | . . . . 5 β’ (π β (FmlaβΟ) β Β¬ π₯ β (((β Sat β )βΟ)βπ)) |
13 | 12 | alrimiv 1930 | . . . 4 β’ (π β (FmlaβΟ) β βπ₯ Β¬ π₯ β (((β Sat β )βΟ)βπ)) |
14 | eq0 4343 | . . . 4 β’ ((((β Sat β )βΟ)βπ) = β β βπ₯ Β¬ π₯ β (((β Sat β )βΟ)βπ)) | |
15 | 13, 14 | sylibr 233 | . . 3 β’ (π β (FmlaβΟ) β (((β Sat β )βΟ)βπ) = β ) |
16 | 1, 15 | eqtrd 2772 | . 2 β’ (π β (FmlaβΟ) β (β Satβ π) = β ) |
17 | prv 34488 | . . . 4 β’ ((β β V β§ π β (FmlaβΟ)) β (β β§π β (β Satβ π) = (β βm Ο))) | |
18 | 8, 17 | mpan 688 | . . 3 β’ (π β (FmlaβΟ) β (β β§π β (β Satβ π) = (β βm Ο))) |
19 | 2 | ne0ii 4337 | . . . . 5 β’ Ο β β |
20 | map0b 8879 | . . . . 5 β’ (Ο β β β (β βm Ο) = β ) | |
21 | 19, 20 | mp1i 13 | . . . 4 β’ (π β (FmlaβΟ) β (β βm Ο) = β ) |
22 | 21 | eqeq2d 2743 | . . 3 β’ (π β (FmlaβΟ) β ((β Satβ π) = (β βm Ο) β (β Satβ π) = β )) |
23 | 18, 22 | bitrd 278 | . 2 β’ (π β (FmlaβΟ) β (β β§π β (β Satβ π) = β )) |
24 | 16, 23 | mpbird 256 | 1 β’ (π β (FmlaβΟ) β β β§π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 βwal 1539 = wceq 1541 β wcel 2106 β wne 2940 Vcvv 3474 β c0 4322 class class class wbr 5148 βΆwf 6539 βcfv 6543 (class class class)co 7411 Οcom 7857 βm cmap 8822 Sat csat 34396 Fmlacfmla 34397 Satβ csate 34398 β§cprv 34399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-ac2 10460 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-ac 10113 df-goel 34400 df-gona 34401 df-goal 34402 df-sat 34403 df-sate 34404 df-fmla 34405 df-prv 34406 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |