Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fmla0disjsuc Structured version   Visualization version   GIF version

Theorem fmla0disjsuc 34684
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 34668 . . . 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 2759 . . 3 (Fmlaβ€˜βˆ…) = {π‘₯ ∣ βˆƒπ‘— ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ π‘₯ = (π‘—βˆˆπ‘”π‘˜)}
43ineq1i 4209 . 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 4300 . . 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 34633 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘—βˆˆπ‘”π‘˜) = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ©)
76eqeq2d 2742 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) β†’ (π‘₯ = (π‘—βˆˆπ‘”π‘˜) ↔ π‘₯ = βŸ¨βˆ…, βŸ¨π‘—, π‘˜βŸ©βŸ©))
8 1n0 8491 . . . . . . . . . . . . . . . . . . . 20 1o β‰  βˆ…
98nesymi 2997 . . . . . . . . . . . . . . . . . . 19 Β¬ βˆ… = 1o
109intnanr 487 . . . . . . . . . . . . . . . . . 18 Β¬ (βˆ… = 1o ∧ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘’, π‘£βŸ©)
11 gonafv 34636 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 ∈ V ∧ 𝑣 ∈ V) β†’ (π‘’βŠΌπ‘”π‘£) = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ©)
1211el2v 3481 . . . . . . . . . . . . . . . . . . . 20 (π‘’βŠΌπ‘”π‘£) = ⟨1o, βŸ¨π‘’, π‘£βŸ©βŸ©
1312eqeq2i 2744 . . . . . . . . . . . . . . . . . . 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 5308 . . . . . . . . . . . . . . . . . . . 20 βˆ… ∈ V
15 opex 5465 . . . . . . . . . . . . . . . . . . . 20 βŸ¨π‘—, π‘˜βŸ© ∈ V
1614, 15opth 5477 . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 14 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
2322adantr 480 . . . . . . . . . . . . 13 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
2423ralrimivw 3149 . . . . . . . . . . . 12 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£))
25 2on0 8485 . . . . . . . . . . . . . . . . . . . . 21 2o β‰  βˆ…
2625nesymi 2997 . . . . . . . . . . . . . . . . . . . 20 Β¬ βˆ… = 2o
2726orci 862 . . . . . . . . . . . . . . . . . . 19 (Β¬ βˆ… = 2o ∨ Β¬ βŸ¨π‘—, π‘˜βŸ© = βŸ¨π‘–, π‘’βŸ©)
2814, 15opth 5477 . . . . . . . . . . . . . . . . . . . . 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 979 . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . 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 34628 . . . . . . . . . . . . . . . . . . . 20 βˆ€π‘”π‘–π‘’ = ⟨2o, βŸ¨π‘–, π‘’βŸ©βŸ©
3534eqeq2i 2744 . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 15 (((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4039adantr 480 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4140adantr 480 . . . . . . . . . . . . 13 (((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) ∧ 𝑖 ∈ Ο‰) β†’ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4241ralrimiva 3145 . . . . . . . . . . . 12 ((((𝑗 ∈ Ο‰ ∧ π‘˜ ∈ Ο‰) ∧ π‘₯ = (π‘—βˆˆπ‘”π‘˜)) ∧ 𝑒 ∈ (Fmlaβ€˜βˆ…)) β†’ βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’)
4324, 42jca 511 . . . . . . . . . . 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 3145 . . . . . . . . . 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 3071 . . . . . . . . . . . . . 14 (βˆ€π‘£ ∈ (Fmlaβ€˜βˆ…) Β¬ π‘₯ = (π‘’βŠΌπ‘”π‘£) ↔ Β¬ βˆƒπ‘£ ∈ (Fmlaβ€˜βˆ…)π‘₯ = (π‘’βŠΌπ‘”π‘£))
46 ralnex 3071 . . . . . . . . . . . . . 14 (βˆ€π‘– ∈ Ο‰ Β¬ π‘₯ = βˆ€π‘”π‘–π‘’ ↔ Β¬ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–π‘’)
4745, 46anbi12i 626 . . . . . . . . . . . . 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 981 . . . . . . . . . . . . 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 3092 . . . . . . . . . . 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 3071 . . . . . . . . . . 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 412 . . . . . . . 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 3154 . . . . . . 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 3147 . . . . . 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 851 . . . . 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 979 . . . . 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 4403 . . 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 2759 . 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 2759 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 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431  Vcvv 3473   ∩ cin 3948  βˆ…c0 4323  βŸ¨cop 4635  β€˜cfv 6544  (class class class)co 7412  Ο‰com 7858  1oc1o 8462  2oc2o 8463  βˆˆπ‘”cgoe 34619  βŠΌπ‘”cgna 34620  βˆ€π‘”cgol 34621  Fmlacfmla 34623
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-map 8825  df-goel 34626  df-gona 34627  df-goal 34628  df-sat 34629  df-fmla 34631
This theorem is referenced by:  satffunlem1lem2  34689
  Copyright terms: Public domain W3C validator