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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 34406 | . . 3 β’ (π β (FmlaβΟ) β (β Satβ π) = (((β Sat β )βΟ)βπ)) | |
2 | peano1 7879 | . . . . . . . . . 10 β’ β β Ο | |
3 | 2 | n0ii 4337 | . . . . . . . . 9 β’ Β¬ Ο = β |
4 | 3 | intnan 488 | . . . . . . . 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 6774 | . . . . . . 7 β’ (π₯:ΟβΆβ β (π₯ = β β§ Ο = β )) | |
7 | 5, 6 | sylnibr 329 | . . . . . 6 β’ (π β (FmlaβΟ) β Β¬ π₯:ΟβΆβ ) |
8 | 0ex 5308 | . . . . . . . 8 β’ β β V | |
9 | 8, 8 | pm3.2i 472 | . . . . . . 7 β’ (β β V β§ β β V) |
10 | satfvel 34403 | . . . . . . 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 1449 | . . . . . 6 β’ ((π β (FmlaβΟ) β§ π₯ β (((β Sat β )βΟ)βπ)) β π₯:ΟβΆβ ) |
12 | 7, 11 | mtand 815 | . . . . 5 β’ (π β (FmlaβΟ) β Β¬ π₯ β (((β Sat β )βΟ)βπ)) |
13 | 12 | alrimiv 1931 | . . . 4 β’ (π β (FmlaβΟ) β βπ₯ Β¬ π₯ β (((β Sat β )βΟ)βπ)) |
14 | eq0 4344 | . . . 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 2773 | . 2 β’ (π β (FmlaβΟ) β (β Satβ π) = β ) |
17 | prv 34419 | . . . 4 β’ ((β β V β§ π β (FmlaβΟ)) β (β β§π β (β Satβ π) = (β βm Ο))) | |
18 | 8, 17 | mpan 689 | . . 3 β’ (π β (FmlaβΟ) β (β β§π β (β Satβ π) = (β βm Ο))) |
19 | 2 | ne0ii 4338 | . . . . 5 β’ Ο β β |
20 | map0b 8877 | . . . . 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 2744 | . . 3 β’ (π β (FmlaβΟ) β ((β Satβ π) = (β βm Ο) β (β Satβ π) = β )) |
23 | 18, 22 | bitrd 279 | . 2 β’ (π β (FmlaβΟ) β (β β§π β (β Satβ π) = β )) |
24 | 16, 23 | mpbird 257 | 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 397 βwal 1540 = wceq 1542 β wcel 2107 β wne 2941 Vcvv 3475 β c0 4323 class class class wbr 5149 βΆwf 6540 βcfv 6544 (class class class)co 7409 Οcom 7855 βm cmap 8820 Sat csat 34327 Fmlacfmla 34328 Satβ csate 34329 β§cprv 34330 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-ac 10111 df-goel 34331 df-gona 34332 df-goal 34333 df-sat 34334 df-sate 34335 df-fmla 34336 df-prv 34337 |
This theorem is referenced by: (None) |
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