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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 5312 | . . 3 ⊢ ∅ ∈ V | |
2 | satefv 35398 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ 𝑉) → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈)) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈)) |
4 | xp0 6179 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
5 | 4 | ineq2i 4224 | . . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅) |
6 | in0 4400 | . . . . . 6 ⊢ ( E ∩ ∅) = ∅ | |
7 | 5, 6 | eqtri 2762 | . . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅ |
8 | 7 | oveq2i 7441 | . . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅) |
9 | 8 | fveq1i 6907 | . . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω) |
10 | 9 | fveq1i 6907 | . 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈) |
11 | 3, 10 | eqtrdi 2790 | 1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ∅c0 4338 E cep 5587 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ωcom 7886 Sat csat 35320 Sat∈ csate 35322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-sate 35328 |
This theorem is referenced by: prv0 35414 |
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