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.

Step	Hyp	Ref	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 <fin “k {n}) ∪ {n}) ↔ (m ∈ (◡k <fin “k {n}) ∨ m ∈ {n}))
4		vex 2862	. . . . . . . . . . . . . 14 ⊢ m ∈ V
5	4	elimak 4259	. . . . . . . . . . . . 13 ⊢ (m ∈ (◡k <fin “k {n}) ↔ ∃t ∈ {n}⟪t, m⟫ ∈ ◡k <fin )
6		vex 2862	. . . . . . . . . . . . . 14 ⊢ n ∈ V
7		opkeq1 4059	. . . . . . . . . . . . . . 15 ⊢ (t = n → ⟪t, m⟫ = ⟪n, m⟫)
8	7	eleq1d 2419	. . . . . . . . . . . . . 14 ⊢ (t = n → (⟪t, m⟫ ∈ ◡k <fin ↔ ⟪n, m⟫ ∈ ◡k <fin ))
9	6, 8	rexsn 3768	. . . . . . . . . . . . 13 ⊢ (∃t ∈ {n}⟪t, m⟫ ∈ ◡k <fin ↔ ⟪n, m⟫ ∈ ◡k <fin )
10	5, 9	bitri 240	. . . . . . . . . . . 12 ⊢ (m ∈ (◡k <fin “k {n}) ↔ ⟪n, m⟫ ∈ ◡k <fin )
11	6, 4	opkelcnvk 4250	. . . . . . . . . . . 12 ⊢ (⟪n, m⟫ ∈ ◡k <fin ↔ ⟪m, n⟫ ∈ <fin )
12	10, 11	bitri 240	. . . . . . . . . . 11 ⊢ (m ∈ (◡k <fin “k {n}) ↔ ⟪m, n⟫ ∈ <fin )
13	4	elsnc 3756	. . . . . . . . . . 11 ⊢ (m ∈ {n} ↔ m = n)
14	12, 13	orbi12i 507	. . . . . . . . . 10 ⊢ ((m ∈ (◡k <fin “k {n}) ∨ m ∈ {n}) ↔ (⟪m, n⟫ ∈ <fin ∨ m = n))
15	3, 14	bitri 240	. . . . . . . . 9 ⊢ (m ∈ ((◡k <fin “k {n}) ∪ {n}) ↔ (⟪m, n⟫ ∈ <fin ∨ m = n))
16	4	elimak 4259	. . . . . . . . . 10 ⊢ (m ∈ ( <fin “k {n}) ↔ ∃t ∈ {n}⟪t, m⟫ ∈ <fin )
17	7	eleq1d 2419	. . . . . . . . . . 11 ⊢ (t = n → (⟪t, m⟫ ∈ <fin ↔ ⟪n, m⟫ ∈ <fin ))
18	6, 17	rexsn 3768	. . . . . . . . . 10 ⊢ (∃t ∈ {n}⟪t, m⟫ ∈ <fin ↔ ⟪n, m⟫ ∈ <fin )
19	16, 18	bitri 240	. . . . . . . . 9 ⊢ (m ∈ ( <fin “k {n}) ↔ ⟪n, m⟫ ∈ <fin )
20	15, 19	orbi12i 507	. . . . . . . 8 ⊢ ((m ∈ ((◡k <fin “k {n}) ∪ {n}) ∨ m ∈ ( <fin “k {n})) ↔ ((⟪m, n⟫ ∈ <fin ∨ m = n) ∨ ⟪n, m⟫ ∈ <fin ))
21		elun 3220	. . . . . . . 8 ⊢ (m ∈ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n})) ↔ (m ∈ ((◡k <fin “k {n}) ∪ {n}) ∨ m ∈ ( <fin “k {n})))
22		df-3or 935	. . . . . . . 8 ⊢ ((⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ) ↔ ((⟪m, n⟫ ∈ <fin ∨ m = n) ∨ ⟪n, m⟫ ∈ <fin ))
23	20, 21, 22	3bitr4i 268	. . . . . . 7 ⊢ (m ∈ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n})) ↔ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))
24	23	abbi2i 2464	. . . . . 6 ⊢ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n})) = {m ∣ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin )}
25	2, 24	uneq12i 3416	. . . . 5 ⊢ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {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 ))}
27	25, 26	eqtri 2373	. . . 4 ⊢ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n}))) = {m ∣ (m = ∅ ∨ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))}
28	27	uneq2i 3415	. . 3 ⊢ ({m ∣ ¬ n ∈ Nn } ∪ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {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 = ∅)
31	30	imbi1i 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 )))
33	31, 32	bitr4i 243	. . . . . 6 ⊢ ((m ≠ ∅ → (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin )) ↔ (m = ∅ ∨ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin )))
34	33	orbi2i 505	. . . . 5 ⊢ ((¬ n ∈ Nn ∨ (m ≠ ∅ → (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))) ↔ (¬ n ∈ Nn ∨ (m = ∅ ∨ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))))
35	29, 34	bitri 240	. . . 4 ⊢ ((n ∈ Nn → (m ≠ ∅ → (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))) ↔ (¬ n ∈ Nn ∨ (m = ∅ ∨ (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin ))))
36	35	abbii 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 )))}
37	1, 28, 36	3eqtr4i 2383	. 2 ⊢ ({m ∣ ¬ n ∈ Nn } ∪ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {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
41	40	cnvkex 4287	. . . . . . 7 ⊢ ◡k <fin ∈ V
42		snex 4111	. . . . . . 7 ⊢ {n} ∈ V
43	41, 42	imakex 4300	. . . . . 6 ⊢ (◡k <fin “k {n}) ∈ V
44	43, 42	unex 4106	. . . . 5 ⊢ ((◡k <fin “k {n}) ∪ {n}) ∈ V
45	40, 42	imakex 4300	. . . . 5 ⊢ ( <fin “k {n}) ∈ V
46	44, 45	unex 4106	. . . 4 ⊢ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n})) ∈ V
47	39, 46	unex 4106	. . 3 ⊢ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n}))) ∈ V
48	38, 47	unex 4106	. 2 ⊢ ({m ∣ ¬ n ∈ Nn } ∪ ({∅} ∪ (((◡k <fin “k {n}) ∪ {n}) ∪ ( <fin “k {n})))) ∈ V
49	37, 48	eqeltrri 2424	1 ⊢ {m ∣ (n ∈ Nn → (m ≠ ∅ → (⟪m, n⟫ ∈ <fin ∨ m = n ∨ ⟪n, m⟫ ∈ <fin )))} ∈ V

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   <_fin cltfin 4433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-addc 4378  df-nnc 4379  df-ltfin 4441

This theorem is referenced by:  ltfintri  4466