Theorem fin23lem23 10248

Description: Lemma for fin23lem22 10249. (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 10247	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
2		ensym 8952	. . . . . 6 ⊢ ((𝑎 ∩ 𝑆) ≈ 𝑖 → 𝑖 ≈ (𝑎 ∩ 𝑆))
3		entr 8955	. . . . . 6 ⊢ (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ 𝑖 ≈ (𝑎 ∩ 𝑆)) → (𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆))
4	2, 3	sylan2 594	. . . . 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 771	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ 𝑆)
7	5, 6	sseldd 3936	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑗 ∈ ω)
8		nnfi 9104	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → 𝑗 ∈ Fin)
9		inss1 4191	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝑆) ⊆ 𝑗
10		ssfi 9109	. . . . . . . . 9 ⊢ ((𝑗 ∈ Fin ∧ (𝑗 ∩ 𝑆) ⊆ 𝑗) → (𝑗 ∩ 𝑆) ∈ Fin)
11	8, 9, 10	sylancl 587	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝑗 ∩ 𝑆) ∈ Fin)
12	7, 11	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ∩ 𝑆) ∈ Fin)
13		simprr 773	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
14	5, 13	sseldd 3936	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
15		nnfi 9104	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
16		inss1 4191	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
17		ssfi 9109	. . . . . . . . 9 ⊢ ((𝑎 ∈ Fin ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ Fin)
18	15, 16, 17	sylancl 587	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎 ∩ 𝑆) ∈ Fin)
19	14, 18	syl 17	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ Fin)
20		nnord 7826	. . . . . . . . . 10 ⊢ (𝑗 ∈ ω → Ord 𝑗)
21		nnord 7826	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → Ord 𝑎)
22		ordtri2or2 6426	. . . . . . . . . 10 ⊢ ((Ord 𝑗 ∧ Ord 𝑎) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
23	20, 21, 22	syl2an 597	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑎 ∈ ω) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
24	7, 14, 23	syl2anc 585	. . . . . . . 8 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (𝑗 ⊆ 𝑎 ∨ 𝑎 ⊆ 𝑗))
25		ssrin 4196	. . . . . . . . 9 ⊢ (𝑗 ⊆ 𝑎 → (𝑗 ∩ 𝑆) ⊆ (𝑎 ∩ 𝑆))
26		ssrin 4196	. . . . . . . . 9 ⊢ (𝑎 ⊆ 𝑗 → (𝑎 ∩ 𝑆) ⊆ (𝑗 ∩ 𝑆))
27	25, 26	orim12i 909	. . . . . . . 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 10246	. . . . . . 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 1374	. . . . . 6 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) ≈ (𝑎 ∩ 𝑆) ↔ (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆)))
31		ordom 7828	. . . . . . 7 ⊢ Ord ω
32		fin23lem24 10244	. . . . . . 7 ⊢ (((Ord ω ∧ 𝑆 ⊆ ω) ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → ((𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆) ↔ 𝑗 = 𝑎))
33	31, 32	mpanl1 701	. . . . . 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 244	. . . 4 ⊢ ((𝑆 ⊆ ω ∧ (𝑗 ∈ 𝑆 ∧ 𝑎 ∈ 𝑆)) → (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
36	35	ralrimivva 3181	. . 3 ⊢ (𝑆 ⊆ ω → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
37	36	ad2antrr 727	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∀𝑗 ∈ 𝑆 ∀𝑎 ∈ 𝑆 (((𝑗 ∩ 𝑆) ≈ 𝑖 ∧ (𝑎 ∩ 𝑆) ≈ 𝑖) → 𝑗 = 𝑎))
38		ineq1 4167	. . . 4 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
39	38	breq1d 5110	. . 3 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
40	39	reu4 3691	. 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 584	1 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350   ∩ cin 3902   ⊆ wss 3903   class class class wbr 5100  Ord word 6324  ωcom 7818   ≈ cen 8892  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899

This theorem is referenced by:  fin23lem22  10249  fin23lem27  10250