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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 5204 | . . 3 ⊢ ∅ ∈ V | |
2 | satefv 32682 | . . 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 6008 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
5 | 4 | ineq2i 4179 | . . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅) |
6 | in0 4338 | . . . . . 6 ⊢ ( E ∩ ∅) = ∅ | |
7 | 5, 6 | eqtri 2843 | . . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅ |
8 | 7 | oveq2i 7160 | . . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅) |
9 | 8 | fveq1i 6664 | . . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω) |
10 | 9 | fveq1i 6664 | . 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈) |
11 | 3, 10 | syl6eq 2871 | 1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∩ cin 3928 ∅c0 4284 E cep 5457 × cxp 5546 ‘cfv 6348 (class class class)co 7149 ωcom 7573 Sat csat 32604 Sat∈ csate 32606 |
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-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-sate 32612 |
This theorem is referenced by: prv0 32698 |
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