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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 5307 | . . 3 ⊢ ∅ ∈ V | |
| 2 | satefv 35419 | . . 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 6178 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
| 5 | 4 | ineq2i 4217 | . . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅) |
| 6 | in0 4395 | . . . . . 6 ⊢ ( E ∩ ∅) = ∅ | |
| 7 | 5, 6 | eqtri 2765 | . . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅ |
| 8 | 7 | oveq2i 7442 | . . . 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 2793 | 1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 E cep 5583 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ωcom 7887 Sat csat 35341 Sat∈ csate 35343 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sate 35349 |
| This theorem is referenced by: prv0 35435 |
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