Theorem prv0 35628

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 35613	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
2		peano1 7833	. . . . . . . . . 10 ⊢ ∅ ∈ ω
3	2	n0ii 4284	. . . . . . . . 9 ⊢ ¬ ω = ∅
4	3	intnan 486	. . . . . . . 8 ⊢ ¬ (𝑥 = ∅ ∧ ω = ∅)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ (𝑥 = ∅ ∧ ω = ∅))
6		f00 6716	. . . . . . 7 ⊢ (𝑥:ω⟶∅ ↔ (𝑥 = ∅ ∧ ω = ∅))
7	5, 6	sylnibr 329	. . . . . 6 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥:ω⟶∅)
8		0ex 5242	. . . . . . . 8 ⊢ ∅ ∈ V
9	8, 8	pm3.2i 470	. . . . . . 7 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
10		satfvel 35610	. . . . . . 7 ⊢ (((∅ ∈ V ∧ ∅ ∈ V) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
11	9, 10	mp3an1 1451	. . . . . 6 ⊢ ((𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
12	7, 11	mtand 816	. . . . 5 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
13	12	alrimiv 1929	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
14		eq0 4291	. . . 4 ⊢ ((((∅ Sat ∅)‘ω)‘𝑈) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
15	13, 14	sylibr 234	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (((∅ Sat ∅)‘ω)‘𝑈) = ∅)
16	1, 15	eqtrd 2772	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = ∅)
17		prv 35626	. . . 4 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ (Fmla‘ω)) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
18	8, 17	mpan 691	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
19	2	ne0ii 4285	. . . . 5 ⊢ ω ≠ ∅
20		map0b 8824	. . . . 5 ⊢ (ω ≠ ∅ → (∅ ↑m ω) = ∅)
21	19, 20	mp1i 13	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ ↑m ω) = ∅)
22	21	eqeq2d 2748	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → ((∅ Sat∈ 𝑈) = (∅ ↑m ω) ↔ (∅ Sat∈ 𝑈) = ∅))
23	18, 22	bitrd 279	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = ∅))
24	16, 23	mpbird 257	1 ⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ∅c0 4274   class class class wbr 5086  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ωcom 7810   ↑_m cmap 8766   Sat csat 35534  Fmlacfmla 35535   Sat_∈ csate 35536  ⊧cprv 35537

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-ac 10029  df-goel 35538  df-gona 35539  df-goal 35540  df-sat 35541  df-sate 35542  df-fmla 35543  df-prv 35544

This theorem is referenced by: (None)