Theorem prv0 33392

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.)

Assertion

Ref	Expression
prv0	⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)

Proof of Theorem prv0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sate0 33377	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
2		peano1 7735	. . . . . . . . . 10 ⊢ ∅ ∈ ω
3	2	n0ii 4270	. . . . . . . . 9 ⊢ ¬ ω = ∅
4	3	intnan 487	. . . . . . . 8 ⊢ ¬ (𝑥 = ∅ ∧ ω = ∅)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ (𝑥 = ∅ ∧ ω = ∅))
6		f00 6656	. . . . . . 7 ⊢ (𝑥:ω⟶∅ ↔ (𝑥 = ∅ ∧ ω = ∅))
7	5, 6	sylnibr 329	. . . . . 6 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥:ω⟶∅)
8		0ex 5231	. . . . . . . 8 ⊢ ∅ ∈ V
9	8, 8	pm3.2i 471	. . . . . . 7 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
10		satfvel 33374	. . . . . . 7 ⊢ (((∅ ∈ V ∧ ∅ ∈ V) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
11	9, 10	mp3an1 1447	. . . . . 6 ⊢ ((𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
12	7, 11	mtand 813	. . . . 5 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
13	12	alrimiv 1930	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
14		eq0 4277	. . . 4 ⊢ ((((∅ Sat ∅)‘ω)‘𝑈) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
15	13, 14	sylibr 233	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (((∅ Sat ∅)‘ω)‘𝑈) = ∅)
16	1, 15	eqtrd 2778	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = ∅)
17		prv 33390	. . . 4 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ (Fmla‘ω)) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
18	8, 17	mpan 687	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
19	2	ne0ii 4271	. . . . 5 ⊢ ω ≠ ∅
20		map0b 8671	. . . . 5 ⊢ (ω ≠ ∅ → (∅ ↑m ω) = ∅)
21	19, 20	mp1i 13	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ ↑m ω) = ∅)
22	21	eqeq2d 2749	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → ((∅ Sat∈ 𝑈) = (∅ ↑m ω) ↔ (∅ Sat∈ 𝑈) = ∅))
23	18, 22	bitrd 278	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = ∅))
24	16, 23	mpbird 256	1 ⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  ∅c0 4256   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ↑_m cmap 8615   Sat csat 33298  Fmlacfmla 33299   Sat_∈ csate 33300  ⊧cprv 33301

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-ac 9872  df-goel 33302  df-gona 33303  df-goal 33304  df-sat 33305  df-sate 33306  df-fmla 33307  df-prv 33308

This theorem is referenced by: (None)