Theorem fin23lem23 9148

Description: Lemma for fin23lem22 9149. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
fin23lem23	⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)

Distinct variable group:   𝑖,𝑗,𝑆

Proof of Theorem fin23lem23

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem26 9147	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
2		ensym 8005	. . . . . 6 ⊢ ((𝑎 ∩ 𝑆) ≈ 𝑖 → 𝑖 ≈ (𝑎 ∩ 𝑆))
3		entr 8008	. . . . . 6 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ 𝑖 ≈ (𝑎 ∩ 𝑆)) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
4	2, 3	sylan2 491	. . . . 5 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
5		simpl 473	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
6		simprl 794	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ 𝑆)
7	5, 6	sseldd 3604	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ ω)
8		nnfi 8153	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → 𝑗 ∈ Fin)
9		inss1 3833	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝑆) ⊆ 𝑗
10		ssfi 8180	. . . . . . . . 9 ⊢ ((𝑗 ∈ Fin ∧ (𝑗 ∩ 𝑆) ⊆ 𝑗) → (𝑗 ∩ 𝑆) ∈ Fin)
11	8, 9, 10	sylancl 694	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝑗 ∩ 𝑆) ∈ Fin)
12	7, 11	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ∩ 𝑆) ∈ Fin)
13		simprr 796	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
14	5, 13	sseldd 3604	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
15		nnfi 8153	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
16		inss1 3833	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
17		ssfi 8180	. . . . . . . . 9 ⊢ ((𝑎 ∈ Fin ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ Fin)
18	15, 16, 17	sylancl 694	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎 ∩ 𝑆) ∈ Fin)
19	14, 18	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ Fin)
20		nnord 7073	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → Ord 𝑗)
21		nnord 7073	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → Ord 𝑎)
22		ordtri2or2 5823	. . . . . . . . . 10 ⊢ ((Ord 𝑗 ∧ Ord 𝑎) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
23	20, 21, 22	syl2an 494	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑎 ∈ ω) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
24	7, 14, 23	syl2anc 693	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
25		ssrin 3838	. . . . . . . . 9 ⊢ (𝑗 ⊆ 𝑎 → (𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆))
26		ssrin 3838	. . . . . . . . 9 ⊢ (𝑎 ⊆ 𝑗 → (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆))
27	25, 26	orim12i 538	. . . . . . . 8 ⊢ ((𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗) → ((𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆) ∨ (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆)))
28	24, 27	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆) ∨ (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆)))
29		fin23lem25 9146	. . . . . . 7 ⊢ (((𝑗 ∩ 𝑆) ∈ Fin ∧ (𝑎 ∩ 𝑆) ∈ Fin ∧ ((𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆) ∨ (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆))) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆)))
30	12, 19, 28, 29	syl3anc 1326	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆)))
31		ordom 7074	. . . . . . 7 ⊢ Ord ω
32		fin23lem24 9144	. . . . . . 7 ⊢ (((Ord ω ∧ 𝑆 ⊆ ω) ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
33	31, 32	mpanl1 716	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
34	30, 33	bitrd 268	. . . . 5 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
35	4, 34	syl5ib 234	. . . 4 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
36	35	ralrimivva 2971	. . 3 ⊢ (𝑆 ⊆ ω → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
37	36	ad2antrr 762	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
38		ineq1 3807	. . . 4 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
39	38	breq1d 4663	. . 3 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
40	39	reu4 3400	. 2 ⊢ (∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖 ∧ ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎)))
41	1, 37, 40	sylanbrc 698	1 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653  Ord word 5722  ωcom 7065   ≈ cen 7952  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  fin23lem22  9149  fin23lem27  9150