Theorem fin23lem23 10317

Description: Lemma for fin23lem22 10318. (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 10316	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
2		ensym 8995	. . . . . 6 ⊢ ((𝑎 ∩ 𝑆) ≈ 𝑖 → 𝑖 ≈ (𝑎 ∩ 𝑆))
3		entr 8998	. . . . . 6 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ 𝑖 ≈ (𝑎 ∩ 𝑆)) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
4	2, 3	sylan2 592	. . . . 5 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
5		simpl 482	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
6		simprl 768	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ 𝑆)
7	5, 6	sseldd 3975	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ ω)
8		nnfi 9163	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → 𝑗 ∈ Fin)
9		inss1 4220	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝑆) ⊆ 𝑗
10		ssfi 9169	. . . . . . . . 9 ⊢ ((𝑗 ∈ Fin ∧ (𝑗 ∩ 𝑆) ⊆ 𝑗) → (𝑗 ∩ 𝑆) ∈ Fin)
11	8, 9, 10	sylancl 585	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝑗 ∩ 𝑆) ∈ Fin)
12	7, 11	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ∩ 𝑆) ∈ Fin)
13		simprr 770	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
14	5, 13	sseldd 3975	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
15		nnfi 9163	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
16		inss1 4220	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
17		ssfi 9169	. . . . . . . . 9 ⊢ ((𝑎 ∈ Fin ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ Fin)
18	15, 16, 17	sylancl 585	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎 ∩ 𝑆) ∈ Fin)
19	14, 18	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ Fin)
20		nnord 7856	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → Ord 𝑗)
21		nnord 7856	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → Ord 𝑎)
22		ordtri2or2 6453	. . . . . . . . . 10 ⊢ ((Ord 𝑗 ∧ Ord 𝑎) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
23	20, 21, 22	syl2an 595	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑎 ∈ ω) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
24	7, 14, 23	syl2anc 583	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
25		ssrin 4225	. . . . . . . . 9 ⊢ (𝑗 ⊆ 𝑎 → (𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆))
26		ssrin 4225	. . . . . . . . 9 ⊢ (𝑎 ⊆ 𝑗 → (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆))
27	25, 26	orim12i 905	. . . . . . . 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 10315	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆)))
31		ordom 7858	. . . . . . 7 ⊢ Ord ω
32		fin23lem24 10313	. . . . . . 7 ⊢ (((Ord ω ∧ 𝑆 ⊆ ω) ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
33	31, 32	mpanl1 697	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
34	30, 33	bitrd 279	. . . . 5 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
35	4, 34	imbitrid 243	. . . 4 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
36	35	ralrimivva 3192	. . 3 ⊢ (𝑆 ⊆ ω → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
37	36	ad2antrr 723	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
38		ineq1 4197	. . . 4 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
39	38	breq1d 5148	. . 3 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
40	39	reu4 3719	. 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 582	1 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  ∃!wreu 3366   ∩ cin 3939   ⊆ wss 3940   class class class wbr 5138  Ord word 6353  ωcom 7848   ≈ cen 8932  Fincfn 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939

This theorem is referenced by:  fin23lem22  10318  fin23lem27  10319