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Mirrors > Home > MPE Home > Th. List > Mathboxes > sate0 | Structured version Visualization version GIF version |
Description: The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.) |
Ref | Expression |
---|---|
sate0 | ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5211 | . . 3 ⊢ ∅ ∈ V | |
2 | satefv 32661 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ 𝑉) → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈)) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈)) |
4 | xp0 6015 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
5 | 4 | ineq2i 4186 | . . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅) |
6 | in0 4345 | . . . . . 6 ⊢ ( E ∩ ∅) = ∅ | |
7 | 5, 6 | eqtri 2844 | . . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅ |
8 | 7 | oveq2i 7167 | . . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅) |
9 | 8 | fveq1i 6671 | . . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω) |
10 | 9 | fveq1i 6671 | . 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈) |
11 | 3, 10 | syl6eq 2872 | 1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ∅c0 4291 E cep 5464 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ωcom 7580 Sat csat 32583 Sat∈ csate 32585 |
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-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-sate 32591 |
This theorem is referenced by: prv0 32677 |
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