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Theorem ennnfonelemim 12425
Description: Lemma for ennnfone 12426. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)
Assertion
Ref Expression
ennnfonelemim (𝐴 β‰ˆ β„• β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—))))
Distinct variable groups:   𝐴,𝑓,𝑗,𝑛   π‘₯,𝐴,𝑦,𝑛   𝑓,π‘˜,𝑗,𝑛   𝑦,𝑗
Allowed substitution hint:   𝐴(π‘˜)
Proof of Theorem ennnfonelemim
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 nn0ennn 10433 . . . 4 β„•0 β‰ˆ β„•
21ensymi 6782 . . 3 β„• β‰ˆ β„•0
3 entr 6784 . . 3 ((𝐴 β‰ˆ β„• ∧ β„• β‰ˆ β„•0) β†’ 𝐴 β‰ˆ β„•0)
42, 3mpan2 425 . 2 (𝐴 β‰ˆ β„• β†’ 𝐴 β‰ˆ β„•0)
5 bren 6747 . . . 4 (𝐴 β‰ˆ β„•0 ↔ βˆƒπ‘” 𝑔:𝐴–1-1-ontoβ†’β„•0)
65biimpi 120 . . 3 (𝐴 β‰ˆ β„•0 β†’ βˆƒπ‘” 𝑔:𝐴–1-1-ontoβ†’β„•0)
7 f1of 5462 . . . . . . . . . . 11 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ 𝑔:π΄βŸΆβ„•0)
87adantr 276 . . . . . . . . . 10 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ 𝑔:π΄βŸΆβ„•0)
9 simprl 529 . . . . . . . . . 10 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ π‘₯ ∈ 𝐴)
108, 9ffvelcdmd 5653 . . . . . . . . 9 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘”β€˜π‘₯) ∈ β„•0)
1110nn0zd 9373 . . . . . . . 8 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘”β€˜π‘₯) ∈ β„€)
12 simprr 531 . . . . . . . . . 10 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ 𝑦 ∈ 𝐴)
138, 12ffvelcdmd 5653 . . . . . . . . 9 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘”β€˜π‘¦) ∈ β„•0)
1413nn0zd 9373 . . . . . . . 8 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘”β€˜π‘¦) ∈ β„€)
15 zdceq 9328 . . . . . . . 8 (((π‘”β€˜π‘₯) ∈ β„€ ∧ (π‘”β€˜π‘¦) ∈ β„€) β†’ DECID (π‘”β€˜π‘₯) = (π‘”β€˜π‘¦))
1611, 14, 15syl2anc 411 . . . . . . 7 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ DECID (π‘”β€˜π‘₯) = (π‘”β€˜π‘¦))
17 dff1o6 5777 . . . . . . . . . . . . 13 (𝑔:𝐴–1-1-ontoβ†’β„•0 ↔ (𝑔 Fn 𝐴 ∧ ran 𝑔 = β„•0 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦)))
1817simp3bi 1014 . . . . . . . . . . . 12 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
1918r19.21bi 2565 . . . . . . . . . . 11 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘¦ ∈ 𝐴 ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
2019r19.21bi 2565 . . . . . . . . . 10 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ π‘₯ ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) β†’ ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
2120anasss 399 . . . . . . . . 9 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
22 fveq2 5516 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (π‘”β€˜π‘₯) = (π‘”β€˜π‘¦))
2321, 22impbid1 142 . . . . . . . 8 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) ↔ π‘₯ = 𝑦))
2423dcbid 838 . . . . . . 7 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (DECID (π‘”β€˜π‘₯) = (π‘”β€˜π‘¦) ↔ DECID π‘₯ = 𝑦))
2516, 24mpbid 147 . . . . . 6 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ DECID π‘₯ = 𝑦)
2625ralrimivva 2559 . . . . 5 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)
27 f1ocnv 5475 . . . . . . 7 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ ◑𝑔:β„•0–1-1-onto→𝐴)
28 f1ofo 5469 . . . . . . 7 (◑𝑔:β„•0–1-1-onto→𝐴 β†’ ◑𝑔:β„•0–onto→𝐴)
2927, 28syl 14 . . . . . 6 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ ◑𝑔:β„•0–onto→𝐴)
30 peano2nn0 9216 . . . . . . . . 9 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•0)
3130adantl 277 . . . . . . . 8 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + 1) ∈ β„•0)
32 elfznn0 10114 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑛) β†’ 𝑗 ∈ β„•0)
3332adantl 277 . . . . . . . . . . . . . 14 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ 𝑗 ∈ β„•0)
3433nn0red 9230 . . . . . . . . . . . . 13 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ 𝑗 ∈ ℝ)
35 elfzle2 10028 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑛) β†’ 𝑗 ≀ 𝑛)
3635adantl 277 . . . . . . . . . . . . . 14 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ 𝑗 ≀ 𝑛)
37 simplr 528 . . . . . . . . . . . . . . 15 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ 𝑛 ∈ β„•0)
38 nn0leltp1 9316 . . . . . . . . . . . . . . 15 ((𝑗 ∈ β„•0 ∧ 𝑛 ∈ β„•0) β†’ (𝑗 ≀ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
3933, 37, 38syl2anc 411 . . . . . . . . . . . . . 14 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ (𝑗 ≀ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
4036, 39mpbid 147 . . . . . . . . . . . . 13 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ 𝑗 < (𝑛 + 1))
4134, 40gtned 8070 . . . . . . . . . . . 12 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ (𝑛 + 1) β‰  𝑗)
4241neneqd 2368 . . . . . . . . . . 11 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ Β¬ (𝑛 + 1) = 𝑗)
43 dff1o6 5777 . . . . . . . . . . . . . . 15 (◑𝑔:β„•0–1-1-onto→𝐴 ↔ (◑𝑔 Fn β„•0 ∧ ran ◑𝑔 = 𝐴 ∧ βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦)))
4427, 43sylib 122 . . . . . . . . . . . . . 14 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ (◑𝑔 Fn β„•0 ∧ ran ◑𝑔 = 𝐴 ∧ βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦)))
4544simp3d 1011 . . . . . . . . . . . . 13 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
4645ad2antrr 488 . . . . . . . . . . . 12 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦))
4731adantr 276 . . . . . . . . . . . . 13 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ (𝑛 + 1) ∈ β„•0)
48 fveqeq2 5525 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑛 + 1) β†’ ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) ↔ (β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘¦)))
49 eqeq1 2184 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑛 + 1) β†’ (π‘₯ = 𝑦 ↔ (𝑛 + 1) = 𝑦))
5048, 49imbi12d 234 . . . . . . . . . . . . . 14 (π‘₯ = (𝑛 + 1) β†’ (((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦) ↔ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘¦) β†’ (𝑛 + 1) = 𝑦)))
51 fveq2 5516 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑗 β†’ (β—‘π‘”β€˜π‘¦) = (β—‘π‘”β€˜π‘—))
5251eqeq2d 2189 . . . . . . . . . . . . . . 15 (𝑦 = 𝑗 β†’ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘¦) ↔ (β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—)))
53 eqeq2 2187 . . . . . . . . . . . . . . 15 (𝑦 = 𝑗 β†’ ((𝑛 + 1) = 𝑦 ↔ (𝑛 + 1) = 𝑗))
5452, 53imbi12d 234 . . . . . . . . . . . . . 14 (𝑦 = 𝑗 β†’ (((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘¦) β†’ (𝑛 + 1) = 𝑦) ↔ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—) β†’ (𝑛 + 1) = 𝑗)))
5550, 54rspc2v 2855 . . . . . . . . . . . . 13 (((𝑛 + 1) ∈ β„•0 ∧ 𝑗 ∈ β„•0) β†’ (βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦) β†’ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—) β†’ (𝑛 + 1) = 𝑗)))
5647, 33, 55syl2anc 411 . . . . . . . . . . . 12 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ (βˆ€π‘₯ ∈ β„•0 βˆ€π‘¦ ∈ β„•0 ((β—‘π‘”β€˜π‘₯) = (β—‘π‘”β€˜π‘¦) β†’ π‘₯ = 𝑦) β†’ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—) β†’ (𝑛 + 1) = 𝑗)))
5746, 56mpd 13 . . . . . . . . . . 11 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ ((β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—) β†’ (𝑛 + 1) = 𝑗))
5842, 57mtod 663 . . . . . . . . . 10 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ Β¬ (β—‘π‘”β€˜(𝑛 + 1)) = (β—‘π‘”β€˜π‘—))
5958neqned 2354 . . . . . . . . 9 (((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) ∧ 𝑗 ∈ (0...𝑛)) β†’ (β—‘π‘”β€˜(𝑛 + 1)) β‰  (β—‘π‘”β€˜π‘—))
6059ralrimiva 2550 . . . . . . . 8 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) β†’ βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜(𝑛 + 1)) β‰  (β—‘π‘”β€˜π‘—))
61 fveq2 5516 . . . . . . . . . . 11 (π‘˜ = (𝑛 + 1) β†’ (β—‘π‘”β€˜π‘˜) = (β—‘π‘”β€˜(𝑛 + 1)))
6261neeq1d 2365 . . . . . . . . . 10 (π‘˜ = (𝑛 + 1) β†’ ((β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—) ↔ (β—‘π‘”β€˜(𝑛 + 1)) β‰  (β—‘π‘”β€˜π‘—)))
6362ralbidv 2477 . . . . . . . . 9 (π‘˜ = (𝑛 + 1) β†’ (βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—) ↔ βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜(𝑛 + 1)) β‰  (β—‘π‘”β€˜π‘—)))
6463rspcev 2842 . . . . . . . 8 (((𝑛 + 1) ∈ β„•0 ∧ βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜(𝑛 + 1)) β‰  (β—‘π‘”β€˜π‘—)) β†’ βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—))
6531, 60, 64syl2anc 411 . . . . . . 7 ((𝑔:𝐴–1-1-ontoβ†’β„•0 ∧ 𝑛 ∈ β„•0) β†’ βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—))
6665ralrimiva 2550 . . . . . 6 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—))
67 cnvexg 5167 . . . . . . . 8 (𝑔 ∈ V β†’ ◑𝑔 ∈ V)
6867elv 2742 . . . . . . 7 ◑𝑔 ∈ V
69 foeq1 5435 . . . . . . . 8 (𝑓 = ◑𝑔 β†’ (𝑓:β„•0–onto→𝐴 ↔ ◑𝑔:β„•0–onto→𝐴))
70 fveq1 5515 . . . . . . . . . . 11 (𝑓 = ◑𝑔 β†’ (π‘“β€˜π‘˜) = (β—‘π‘”β€˜π‘˜))
71 fveq1 5515 . . . . . . . . . . 11 (𝑓 = ◑𝑔 β†’ (π‘“β€˜π‘—) = (β—‘π‘”β€˜π‘—))
7270, 71neeq12d 2367 . . . . . . . . . 10 (𝑓 = ◑𝑔 β†’ ((π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—) ↔ (β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—)))
7372rexralbidv 2503 . . . . . . . . 9 (𝑓 = ◑𝑔 β†’ (βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—) ↔ βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—)))
7473ralbidv 2477 . . . . . . . 8 (𝑓 = ◑𝑔 β†’ (βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—) ↔ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—)))
7569, 74anbi12d 473 . . . . . . 7 (𝑓 = ◑𝑔 β†’ ((𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—)) ↔ (◑𝑔:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—))))
7668, 75spcev 2833 . . . . . 6 ((◑𝑔:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(β—‘π‘”β€˜π‘˜) β‰  (β—‘π‘”β€˜π‘—)) β†’ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—)))
7729, 66, 76syl2anc 411 . . . . 5 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—)))
7826, 77jca 306 . . . 4 (𝑔:𝐴–1-1-ontoβ†’β„•0 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—))))
7978adantl 277 . . 3 ((𝐴 β‰ˆ β„•0 ∧ 𝑔:𝐴–1-1-ontoβ†’β„•0) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—))))
806, 79exlimddv 1898 . 2 (𝐴 β‰ˆ β„•0 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—))))
814, 80syl 14 1 (𝐴 β‰ˆ β„• β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:β„•0–onto→𝐴 ∧ βˆ€π‘› ∈ β„•0 βˆƒπ‘˜ ∈ β„•0 βˆ€π‘— ∈ (0...𝑛)(π‘“β€˜π‘˜) β‰  (π‘“β€˜π‘—))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 834   ∧ w3a 978   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148   β‰  wne 2347  βˆ€wral 2455  βˆƒwrex 2456  Vcvv 2738   class class class wbr 4004  β—‘ccnv 4626  ran crn 4628   Fn wfn 5212  βŸΆwf 5213  β€“ontoβ†’wfo 5215  β€“1-1-ontoβ†’wf1o 5216  β€˜cfv 5217  (class class class)co 5875   β‰ˆ cen 6738  0cc0 7811  1c1 7812   + caddc 7814   < clt 7992   ≀ cle 7993  β„•cn 8919  β„•0cn0 9176  β„€cz 9253  ...cfz 10008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-er 6535  df-en 6741  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009
This theorem is referenced by:  ennnfone  12426
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