Theorem fin23lem23 10283

Description: Lemma for fin23lem22 10284. (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 10282	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
2		ensym 8984	. . . . . 6 ⊢ ((𝑎 ∩ 𝑆) ≈ 𝑖 → 𝑖 ≈ (𝑎 ∩ 𝑆))
3		entr 8987	. . . . . 6 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ 𝑖 ≈ (𝑎 ∩ 𝑆)) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
4	2, 3	sylan2 602	. . . . 5 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
5		simpl 486	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
6		simprl 780	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ 𝑆)
7	5, 6	sseldd 3937	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ ω)
8		nnfi 9136	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → 𝑗 ∈ Fin)
9		inss1 4188	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝑆) ⊆ 𝑗
10		ssfi 9141	. . . . . . . . 9 ⊢ ((𝑗 ∈ Fin ∧ (𝑗 ∩ 𝑆) ⊆ 𝑗) → (𝑗 ∩ 𝑆) ∈ Fin)
11	8, 9, 10	sylancl 595	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝑗 ∩ 𝑆) ∈ Fin)
12	7, 11	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ∩ 𝑆) ∈ Fin)
13		simprr 782	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
14	5, 13	sseldd 3937	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
15		nnfi 9136	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
16		inss1 4188	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
17		ssfi 9141	. . . . . . . . 9 ⊢ ((𝑎 ∈ Fin ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ Fin)
18	15, 16, 17	sylancl 595	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎 ∩ 𝑆) ∈ Fin)
19	14, 18	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ Fin)
20		nnord 7854	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → Ord 𝑗)
21		nnord 7854	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → Ord 𝑎)
22		ordtri2or2 6447	. . . . . . . . . 10 ⊢ ((Ord 𝑗 ∧ Ord 𝑎) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
23	20, 21, 22	syl2an 605	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑎 ∈ ω) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
24	7, 14, 23	syl2anc 593	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
25		ssrin 4193	. . . . . . . . 9 ⊢ (𝑗 ⊆ 𝑎 → (𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆))
26		ssrin 4193	. . . . . . . . 9 ⊢ (𝑎 ⊆ 𝑗 → (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆))
27	25, 26	orim12i 919	. . . . . . . 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 10281	. . . . . . 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 1390	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆)))
31		ordom 7856	. . . . . . 7 ⊢ Ord ω
32		fin23lem24 10279	. . . . . . 7 ⊢ (((Ord ω ∧ 𝑆 ⊆ ω) ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
33	31, 32	mpanl1 710	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
34	30, 33	bitrd 281	. . . . 5 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
35	4, 34	imbitrid 246	. . . 4 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
36	35	ralrimivva 3205	. . 3 ⊢ (𝑆 ⊆ ω → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
37	36	ad2antrr 736	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
38		ineq1 4165	. . . 4 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
39	38	breq1d 5110	. . 3 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
40	39	reu4 3694	. 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 592	1 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  ∃!wreu 3365   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5100  Ord word 6345  ωcom 7846   ≈ cen 8924  Fincfn 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931

This theorem is referenced by:  fin23lem22  10284  fin23lem27  10285