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Theorem fmla0disjsuc 34458
Description: The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)
Assertion
Ref Expression
fmla0disjsuc ((Fmlaβ€˜βˆ…) ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)}) = βˆ…
Distinct variable group:   𝑒,𝑖,𝑣,π‘₯

Proof of Theorem fmla0disjsuc
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmla0 34442 . . . 4 (Fmlaβ€˜βˆ…) = {π‘₯ ∈ V ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)}
2 rabab 3502 . . . 4 {π‘₯ ∈ V ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)} = {π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)}
31, 2eqtri 2760 . . 3 (Fmlaβ€˜βˆ…) = {π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)}
43ineq1i 4208 . 2 ((Fmlaβ€˜βˆ…) ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)}) = ({π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)} ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)})
5 inab 4299 . . 3 ({π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)} ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)}) = {π‘₯ ∣ (βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∧ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))}
6 goel 34407 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘—βˆˆπ‘”π‘˜) = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ©)
76eqeq2d 2743 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘₯ = (π‘—βˆˆπ‘”π‘˜) ↔ π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ©))
8 1n0 8490 . . . . . . . . . . . . . . . . . . . 20 1o β‰  βˆ…
98nesymi 2998 . . . . . . . . . . . . . . . . . . 19 Β¬ βˆ… = 1o
109intnanr 488 . . . . . . . . . . . . . . . . . 18 Β¬ (βˆ… = 1o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘’, π‘£βŸ©)
11 gonafv 34410 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 ∈ V ∧ 𝑣 ∈ V) β†’ (π‘’βŠΌπ‘”π‘£) = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ©)
1211el2v 3482 . . . . . . . . . . . . . . . . . . . 20 (π‘’βŠΌπ‘”π‘£) = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ©
1312eqeq2i 2745 . . . . . . . . . . . . . . . . . . 19 (βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = (π‘’βŠΌπ‘”π‘£) ↔ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ©)
14 0ex 5307 . . . . . . . . . . . . . . . . . . . 20 βˆ… ∈ V
15 opex 5464 . . . . . . . . . . . . . . . . . . . 20 βŸ¨π‘—, π‘˜βŸ© ∈ V
1614, 15opth 5476 . . . . . . . . . . . . . . . . . . 19 (βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ© ↔ (βˆ… = 1o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘’, π‘£βŸ©))
1713, 16bitri 274 . . . . . . . . . . . . . . . . . 18 (βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = (π‘’βŠΌπ‘”π‘£) ↔ (βˆ… = 1o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘’, π‘£βŸ©))
1810, 17mtbir 322 . . . . . . . . . . . . . . . . 17 Β¬ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = (π‘’βŠΌπ‘”π‘£)
19 eqeq1 2736 . . . . . . . . . . . . . . . . 17 (π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© β†’ (π‘₯ = (π‘’βŠΌπ‘”π‘£) ↔ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = (π‘’βŠΌπ‘”π‘£)))
2018, 19mtbiri 326 . . . . . . . . . . . . . . . 16 (π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
217, 20syl6bi 252 . . . . . . . . . . . . . . 15 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘₯ = (π‘—βˆˆπ‘”π‘˜) β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£)))
2221imp 407 . . . . . . . . . . . . . 14 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
2322adantr 481 . . . . . . . . . . . . 13 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
2423ralrimivw 3150 . . . . . . . . . . . 12 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
25 2on0 8484 . . . . . . . . . . . . . . . . . . . . 21 2o β‰  βˆ…
2625nesymi 2998 . . . . . . . . . . . . . . . . . . . 20 Β¬ βˆ… = 2o
2726orci 863 . . . . . . . . . . . . . . . . . . 19 (Β¬ βˆ… = 2o ∨ Β¬ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©)
2814, 15opth 5476 . . . . . . . . . . . . . . . . . . . . 21 (βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ© ↔ (βˆ… = 2o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©))
2928notbii 319 . . . . . . . . . . . . . . . . . . . 20 (Β¬ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ© ↔ Β¬ (βˆ… = 2o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©))
30 ianor 980 . . . . . . . . . . . . . . . . . . . 20 (Β¬ (βˆ… = 2o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©) ↔ (Β¬ βˆ… = 2o ∨ Β¬ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©))
3129, 30bitri 274 . . . . . . . . . . . . . . . . . . 19 (Β¬ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ© ↔ (Β¬ βˆ… = 2o ∨ Β¬ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©))
3227, 31mpbir 230 . . . . . . . . . . . . . . . . . 18 Β¬ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ©
33 eqeq1 2736 . . . . . . . . . . . . . . . . . . 19 (π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© β†’ (π‘₯ = βˆ€π‘”π‘–π‘’ ↔ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = βˆ€π‘”π‘–π‘’))
34 df-goal 34402 . . . . . . . . . . . . . . . . . . . 20 βˆ€π‘”π‘–π‘’ = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ©
3534eqeq2i 2745 . . . . . . . . . . . . . . . . . . 19 (βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = βˆ€π‘”π‘–π‘’ ↔ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ©)
3633, 35bitrdi 286 . . . . . . . . . . . . . . . . . 18 (π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© β†’ (π‘₯ = βˆ€π‘”π‘–π‘’ ↔ βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ©))
3732, 36mtbiri 326 . . . . . . . . . . . . . . . . 17 (π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ© β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
387, 37syl6bi 252 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘₯ = (π‘—βˆˆπ‘”π‘˜) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’))
3938imp 407 . . . . . . . . . . . . . . 15 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4039adantr 481 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4140adantr 481 . . . . . . . . . . . . 13 (((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) ∧ 𝑖 ∈ Ο‰) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4241ralrimiva 3146 . . . . . . . . . . . 12 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4324, 42jca 512 . . . . . . . . . . 11 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ (βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’))
4443ralrimiva 3146 . . . . . . . . . 10 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ βˆ€π‘’ ∈ (Fmlaβ€˜βˆ…)(βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’))
45 ralnex 3072 . . . . . . . . . . . . . 14 (βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ↔ Β¬ βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£))
46 ralnex 3072 . . . . . . . . . . . . . 14 (βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’ ↔ Β¬ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)
4745, 46anbi12i 627 . . . . . . . . . . . . 13 ((βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ (Β¬ βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ Β¬ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
48 ioran 982 . . . . . . . . . . . . 13 (Β¬ (βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ (Β¬ βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ Β¬ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
4947, 48bitr4i 277 . . . . . . . . . . . 12 ((βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ Β¬ (βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
5049ralbii 3093 . . . . . . . . . . 11 (βˆ€π‘’ ∈ (Fmlaβ€˜βˆ…)(βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ βˆ€π‘’ ∈ (Fmlaβ€˜βˆ…) Β¬ (βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
51 ralnex 3072 . . . . . . . . . . 11 (βˆ€π‘’ ∈ (Fmlaβ€˜βˆ…) Β¬ (βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
5250, 51bitri 274 . . . . . . . . . 10 (βˆ€π‘’ ∈ (Fmlaβ€˜βˆ…)(βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ∧ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’) ↔ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
5344, 52sylib 217 . . . . . . . . 9 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
5453ex 413 . . . . . . . 8 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘₯ = (π‘—βˆˆπ‘”π‘˜) β†’ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)))
5554rexlimdva 3155 . . . . . . 7 (𝑗 ∈ Ο‰ β†’ (βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) β†’ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)))
5655rexlimiv 3148 . . . . . 6 (βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) β†’ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
5756imori 852 . . . . 5 (Β¬ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∨ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
58 ianor 980 . . . . 5 (Β¬ (βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∧ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)) ↔ (Β¬ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∨ Β¬ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)))
5957, 58mpbir 230 . . . 4 Β¬ (βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∧ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))
6059abf 4402 . . 3 {π‘₯ ∣ (βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜) ∧ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’))} = βˆ…
615, 60eqtri 2760 . 2 ({π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)} ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)}) = βˆ…
624, 61eqtri 2760 1 ((Fmlaβ€˜βˆ…) ∩ {π‘₯ ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜βˆ…)(βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)}) = βˆ…
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947  βˆ…c0 4322  βŸ¨cop 4634  β€˜cfv 6543  (class class class)co 7411  Ο‰com 7857  1oc1o 8461  2oc2o 8462  βˆˆπ‘”cgoe 34393  βŠΌπ‘”cgna 34394  βˆ€π‘”cgol 34395  Fmlacfmla 34397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-map 8824  df-goel 34400  df-gona 34401  df-goal 34402  df-sat 34403  df-fmla 34405
This theorem is referenced by:  satffunlem1lem2  34463
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