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Mirrors > Home > MPE Home > Th. List > Mathboxes > prv0 | Structured version Visualization version GIF version |
Description: Every wff encoded as 𝑈 is true in an "empty model" (𝑀 = ∅). Since ⊧ 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 ⊧ should not be interpreted as the empty model, because ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.) |
Ref | Expression |
---|---|
prv0 | ⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sate0 32662 | . . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) | |
2 | peano1 7601 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
3 | 2 | n0ii 4302 | . . . . . . . . 9 ⊢ ¬ ω = ∅ |
4 | 3 | intnan 489 | . . . . . . . 8 ⊢ ¬ (𝑥 = ∅ ∧ ω = ∅) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ (𝑥 = ∅ ∧ ω = ∅)) |
6 | f00 6561 | . . . . . . 7 ⊢ (𝑥:ω⟶∅ ↔ (𝑥 = ∅ ∧ ω = ∅)) | |
7 | 5, 6 | sylnibr 331 | . . . . . 6 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥:ω⟶∅) |
8 | 0ex 5211 | . . . . . . . 8 ⊢ ∅ ∈ V | |
9 | 8, 8 | pm3.2i 473 | . . . . . . 7 ⊢ (∅ ∈ V ∧ ∅ ∈ V) |
10 | satfvel 32659 | . . . . . . 7 ⊢ (((∅ ∈ V ∧ ∅ ∈ V) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅) | |
11 | 9, 10 | mp3an1 1444 | . . . . . 6 ⊢ ((𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅) |
12 | 7, 11 | mtand 814 | . . . . 5 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) |
13 | 12 | alrimiv 1928 | . . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) |
14 | eq0 4308 | . . . 4 ⊢ ((((∅ Sat ∅)‘ω)‘𝑈) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) | |
15 | 13, 14 | sylibr 236 | . . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (((∅ Sat ∅)‘ω)‘𝑈) = ∅) |
16 | 1, 15 | eqtrd 2856 | . 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = ∅) |
17 | prv 32675 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ (Fmla‘ω)) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω))) | |
18 | 8, 17 | mpan 688 | . . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω))) |
19 | 2 | ne0ii 4303 | . . . . 5 ⊢ ω ≠ ∅ |
20 | map0b 8447 | . . . . 5 ⊢ (ω ≠ ∅ → (∅ ↑m ω) = ∅) | |
21 | 19, 20 | mp1i 13 | . . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ ↑m ω) = ∅) |
22 | 21 | eqeq2d 2832 | . . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → ((∅ Sat∈ 𝑈) = (∅ ↑m ω) ↔ (∅ Sat∈ 𝑈) = ∅)) |
23 | 18, 22 | bitrd 281 | . 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = ∅)) |
24 | 16, 23 | mpbird 259 | 1 ⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ωcom 7580 ↑m cmap 8406 Sat csat 32583 Fmlacfmla 32584 Sat∈ csate 32585 ⊧cprv 32586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-ac2 9885 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-card 9368 df-ac 9542 df-goel 32587 df-gona 32588 df-goal 32589 df-sat 32590 df-sate 32591 df-fmla 32592 df-prv 32593 |
This theorem is referenced by: (None) |
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