Theorem prv0 32701

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 32686	. . 3 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
2		peano1 7598	. . . . . . . . . 10 ⊢ ∅ ∈ ω
3	2	n0ii 4299	. . . . . . . . 9 ⊢ ¬ ω = ∅
4	3	intnan 489	. . . . . . . 8 ⊢ ¬ (𝑥 = ∅ ∧ ω = ∅)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ (𝑥 = ∅ ∧ ω = ∅))
6		f00 6558	. . . . . . 7 ⊢ (𝑥:ω⟶∅ ↔ (𝑥 = ∅ ∧ ω = ∅))
7	5, 6	sylnibr 331	. . . . . 6 ⊢ (𝑈 ∈ (Fmla‘ω) → ¬ 𝑥:ω⟶∅)
8		0ex 5208	. . . . . . . 8 ⊢ ∅ ∈ V
9	8, 8	pm3.2i 473	. . . . . . 7 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
10		satfvel 32683	. . . . . . 7 ⊢ (((∅ ∈ V ∧ ∅ ∈ V) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈)) → 𝑥:ω⟶∅)
11	9, 10	mp3an1 1443	. . . . . 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 1927	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → ∀𝑥 ¬ 𝑥 ∈ (((∅ Sat ∅)‘ω)‘𝑈))
14		eq0 4305	. . . 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 2855	. 2 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ Sat∈ 𝑈) = ∅)
17		prv 32699	. . . 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 4300	. . . . 5 ⊢ ω ≠ ∅
20		map0b 8444	. . . . 5 ⊢ (ω ≠ ∅ → (∅ ↑m ω) = ∅)
21	19, 20	mp1i 13	. . . 4 ⊢ (𝑈 ∈ (Fmla‘ω) → (∅ ↑m ω) = ∅)
22	21	eqeq2d 2831	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534   = wceq 1536   ∈ wcel 2113   ≠ wne 3015  Vcvv 3493  ∅c0 4288   class class class wbr 5063  ⟶wf 6348  ‘cfv 6352  (class class class)co 7153  ωcom 7577   ↑_m cmap 8403   Sat csat 32607  Fmlacfmla 32608   Sat_∈ csate 32609  ⊧cprv 32610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458  ax-inf2 9101  ax-ac2 9882

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4836  df-int 4874  df-iun 4918  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5457  df-eprel 5462  df-po 5471  df-so 5472  df-fr 5511  df-se 5512  df-we 5513  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7111  df-ov 7156  df-oprab 7157  df-mpo 7158  df-om 7578  df-1st 7686  df-2nd 7687  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-2o 8100  df-er 8286  df-map 8405  df-en 8507  df-dom 8508  df-sdom 8509  df-card 9365  df-ac 9539  df-goel 32611  df-gona 32612  df-goal 32613  df-sat 32614  df-sate 32615  df-fmla 32616  df-prv 32617

This theorem is referenced by: (None)