Theorem prv0 35624

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 35609	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
2		peano1 7831	. . . . . . . . . 10 ⊢ ∅ ∈ ω
3	2	n0ii 4295	. . . . . . . . 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 5252	. . . . . . . 8 ⊢ ∅ ∈ V
9	8, 8	pm3.2i 470	. . . . . . 7 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
10		satfvel 35606	. . . . . . 7 ⊢ (((∅ ∈ V ∧ ∅ ∈ V) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
11	9, 10	mp3an1 1450	. . . . . 6 ⊢ ((𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
12	7, 11	mtand 815	. . . . 5 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
13	12	alrimiv 1928	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
14		eq0 4302	. . . 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 2771	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = ∅)
17		prv 35622	. . . 4 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ (Fmla‘ω)) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
18	8, 17	mpan 690	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅⊧𝑈 ↔ (∅ Sat∈ 𝑈) = (∅ ↑m ω)))
19	2	ne0ii 4296	. . . . 5 ⊢ ω ≠ ∅
20		map0b 8821	. . . . 5 ⊢ (ω ≠ ∅ → (∅ ↑m ω) = ∅)
21	19, 20	mp1i 13	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ ↑m ω) = ∅)
22	21	eqeq2d 2747	. . 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 1539   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ∅c0 4285   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ωcom 7808   ↑_m cmap 8763   Sat csat 35530  Fmlacfmla 35531   Sat_∈ csate 35532  ⊧cprv 35533

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-ac2 10373

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-ac 10026  df-goel 35534  df-gona 35535  df-goal 35536  df-sat 35537  df-sate 35538  df-fmla 35539  df-prv 35540

This theorem is referenced by: (None)