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Theorem ltfintrilem1 4465
 Description: Lemma for ltfintri 4466. Establish stratification for induction. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
ltfintrilem1 {m (n Nn → (m → (⟪m, n <fin m = n n, m <fin )))} V
Distinct variable group:   m,n

Proof of Theorem ltfintrilem1
Dummy variable t is distinct from all other variables.
StepHypRef Expression
1 unab 3521 . . 3 ({m ¬ n Nn } ∪ {m (m = (⟪m, n <fin m = n n, m <fin ))}) = {m n Nn (m = (⟪m, n <fin m = n n, m <fin )))}
2 df-sn 3741 . . . . . 6 {} = {m m = }
3 elun 3220 . . . . . . . . . 10 (m ((k <fink {n}) ∪ {n}) ↔ (m (k <fink {n}) m {n}))
4 vex 2862 . . . . . . . . . . . . . 14 m V
54elimak 4259 . . . . . . . . . . . . 13 (m (k <fink {n}) ↔ t {n}⟪t, m k <fin )
6 vex 2862 . . . . . . . . . . . . . 14 n V
7 opkeq1 4059 . . . . . . . . . . . . . . 15 (t = n → ⟪t, m⟫ = ⟪n, m⟫)
87eleq1d 2419 . . . . . . . . . . . . . 14 (t = n → (⟪t, m k <fin ↔ ⟪n, m k <fin ))
96, 8rexsn 3768 . . . . . . . . . . . . 13 (t {n}⟪t, m k <fin ↔ ⟪n, m k <fin )
105, 9bitri 240 . . . . . . . . . . . 12 (m (k <fink {n}) ↔ ⟪n, m k <fin )
116, 4opkelcnvk 4250 . . . . . . . . . . . 12 (⟪n, m k <fin ↔ ⟪m, n <fin )
1210, 11bitri 240 . . . . . . . . . . 11 (m (k <fink {n}) ↔ ⟪m, n <fin )
134elsnc 3756 . . . . . . . . . . 11 (m {n} ↔ m = n)
1412, 13orbi12i 507 . . . . . . . . . 10 ((m (k <fink {n}) m {n}) ↔ (⟪m, n <fin m = n))
153, 14bitri 240 . . . . . . . . 9 (m ((k <fink {n}) ∪ {n}) ↔ (⟪m, n <fin m = n))
164elimak 4259 . . . . . . . . . 10 (m ( <fink {n}) ↔ t {n}⟪t, m <fin )
177eleq1d 2419 . . . . . . . . . . 11 (t = n → (⟪t, m <fin ↔ ⟪n, m <fin ))
186, 17rexsn 3768 . . . . . . . . . 10 (t {n}⟪t, m <fin ↔ ⟪n, m <fin )
1916, 18bitri 240 . . . . . . . . 9 (m ( <fink {n}) ↔ ⟪n, m <fin )
2015, 19orbi12i 507 . . . . . . . 8 ((m ((k <fink {n}) ∪ {n}) m ( <fink {n})) ↔ ((⟪m, n <fin m = n) n, m <fin ))
21 elun 3220 . . . . . . . 8 (m (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})) ↔ (m ((k <fink {n}) ∪ {n}) m ( <fink {n})))
22 df-3or 935 . . . . . . . 8 ((⟪m, n <fin m = n n, m <fin ) ↔ ((⟪m, n <fin m = n) n, m <fin ))
2320, 21, 223bitr4i 268 . . . . . . 7 (m (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})) ↔ (⟪m, n <fin m = n n, m <fin ))
2423abbi2i 2464 . . . . . 6 (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})) = {m (⟪m, n <fin m = n n, m <fin )}
252, 24uneq12i 3416 . . . . 5 ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n}))) = ({m m = } ∪ {m (⟪m, n <fin m = n n, m <fin )})
26 unab 3521 . . . . 5 ({m m = } ∪ {m (⟪m, n <fin m = n n, m <fin )}) = {m (m = (⟪m, n <fin m = n n, m <fin ))}
2725, 26eqtri 2373 . . . 4 ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n}))) = {m (m = (⟪m, n <fin m = n n, m <fin ))}
2827uneq2i 3415 . . 3 ({m ¬ n Nn } ∪ ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})))) = ({m ¬ n Nn } ∪ {m (m = (⟪m, n <fin m = n n, m <fin ))})
29 imor 401 . . . . 5 ((n Nn → (m → (⟪m, n <fin m = n n, m <fin ))) ↔ (¬ n Nn (m → (⟪m, n <fin m = n n, m <fin ))))
30 df-ne 2518 . . . . . . . 8 (m ↔ ¬ m = )
3130imbi1i 315 . . . . . . 7 ((m → (⟪m, n <fin m = n n, m <fin )) ↔ (¬ m = → (⟪m, n <fin m = n n, m <fin )))
32 df-or 359 . . . . . . 7 ((m = (⟪m, n <fin m = n n, m <fin )) ↔ (¬ m = → (⟪m, n <fin m = n n, m <fin )))
3331, 32bitr4i 243 . . . . . 6 ((m → (⟪m, n <fin m = n n, m <fin )) ↔ (m = (⟪m, n <fin m = n n, m <fin )))
3433orbi2i 505 . . . . 5 ((¬ n Nn (m → (⟪m, n <fin m = n n, m <fin ))) ↔ (¬ n Nn (m = (⟪m, n <fin m = n n, m <fin ))))
3529, 34bitri 240 . . . 4 ((n Nn → (m → (⟪m, n <fin m = n n, m <fin ))) ↔ (¬ n Nn (m = (⟪m, n <fin m = n n, m <fin ))))
3635abbii 2465 . . 3 {m (n Nn → (m → (⟪m, n <fin m = n n, m <fin )))} = {m n Nn (m = (⟪m, n <fin m = n n, m <fin )))}
371, 28, 363eqtr4i 2383 . 2 ({m ¬ n Nn } ∪ ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})))) = {m (n Nn → (m → (⟪m, n <fin m = n n, m <fin )))}
38 abexv 4324 . . 3 {m ¬ n Nn } V
39 snex 4111 . . . 4 {} V
40 ltfinex 4464 . . . . . . . 8 <fin V
4140cnvkex 4287 . . . . . . 7 k <fin V
42 snex 4111 . . . . . . 7 {n} V
4341, 42imakex 4300 . . . . . 6 (k <fink {n}) V
4443, 42unex 4106 . . . . 5 ((k <fink {n}) ∪ {n}) V
4540, 42imakex 4300 . . . . 5 ( <fink {n}) V
4644, 45unex 4106 . . . 4 (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})) V
4739, 46unex 4106 . . 3 ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n}))) V
4838, 47unex 4106 . 2 ({m ¬ n Nn } ∪ ({} ∪ (((k <fink {n}) ∪ {n}) ∪ ( <fink {n})))) V
4937, 48eqeltrri 2424 1 {m (n Nn → (m → (⟪m, n <fin m = n n, m <fin )))} V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∨ w3o 933   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∪ cun 3207  ∅c0 3550  {csn 3737  ⟪copk 4057  ◡kccnvk 4175   “k cimak 4179   Nn cnnc 4373
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