Theorem cat1lem 18058

Description: The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 18059. (Contributed by Zhi Wang, 15-Sep-2024.)

Hypotheses

Ref	Expression
cat1lem.1	⊢ 𝐶 = (SetCat‘𝑈)
cat1lem.2	⊢ (𝜑 → 𝑈 ∈ 𝑉)
cat1lem.3	⊢ 𝐵 = (Base‘𝐶)
cat1lem.4	⊢ 𝐻 = (Hom ‘𝐶)
cat1lem.5	⊢ (𝜑 → ∅ ∈ 𝑈)
cat1lem.6	⊢ (𝜑 → 𝑌 ∈ 𝑈)
cat1lem.7	⊢ (𝜑 → ∅ ≠ 𝑌)

Assertion

Ref	Expression
cat1lem	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Distinct variable groups:   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐻,𝑥,𝑦,𝑧   𝑤,𝑌

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝑈(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem cat1lem

Step	Hyp	Ref	Expression
1		cat1lem.5	. . 3 ⊢ (𝜑 → ∅ ∈ 𝑈)
2		cat1lem.1	. . . . 5 ⊢ 𝐶 = (SetCat‘𝑈)
3		cat1lem.2	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	2, 3	setcbas 18040	. . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
5		cat1lem.3	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2782	. . 3 ⊢ (𝜑 → 𝑈 = 𝐵)
7	1, 6	eleqtrd 2830	. 2 ⊢ (𝜑 → ∅ ∈ 𝐵)
8		cat1lem.6	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
9	8, 6	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		f0 6741	. . . . 5 ⊢ ∅:∅⟶∅
11		cat1lem.4	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
12	2, 3, 11, 1, 1	elsetchom 18043	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅))
13	10, 12	mpbiri 258	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻∅))
14		f0 6741	. . . . 5 ⊢ ∅:∅⟶𝑌
15	2, 3, 11, 1, 8	elsetchom 18043	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻𝑌) ↔ ∅:∅⟶𝑌))
16	14, 15	mpbiri 258	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻𝑌))
17		inelcm 4428	. . . 4 ⊢ ((∅ ∈ (∅𝐻∅) ∧ ∅ ∈ (∅𝐻𝑌)) → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
18	13, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
19		cat1lem.7	. . . . 5 ⊢ (𝜑 → ∅ ≠ 𝑌)
20	19	neneqd 2930	. . . 4 ⊢ (𝜑 → ¬ ∅ = 𝑌)
21	20	intnand 488	. . 3 ⊢ (𝜑 → ¬ (∅ = ∅ ∧ ∅ = 𝑌))
22		oveq1 7394	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧𝐻𝑤) = (∅𝐻𝑤))
23	22	ineq2d 4183	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑤)))
24	23	neeq1d 2984	. . . . 5 ⊢ (𝑧 = ∅ → (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅))
25		eqeq2 2741	. . . . . . 7 ⊢ (𝑧 = ∅ → (∅ = 𝑧 ↔ ∅ = ∅))
26	25	anbi1d 631	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅ = 𝑧 ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑤)))
27	26	notbid 318	. . . . 5 ⊢ (𝑧 = ∅ → (¬ (∅ = 𝑧 ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑤)))
28	24, 27	anbi12d 632	. . . 4 ⊢ (𝑧 = ∅ → ((((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤))))
29		oveq2 7395	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅𝐻𝑤) = (∅𝐻𝑌))
30	29	ineq2d 4183	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅𝐻∅) ∩ (∅𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑌)))
31	30	neeq1d 2984	. . . . 5 ⊢ (𝑤 = 𝑌 → (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅))
32		eqeq2 2741	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅ = 𝑤 ↔ ∅ = 𝑌))
33	32	anbi2d 630	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅ = ∅ ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑌)))
34	33	notbid 318	. . . . 5 ⊢ (𝑤 = 𝑌 → (¬ (∅ = ∅ ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑌)))
35	31, 34	anbi12d 632	. . . 4 ⊢ (𝑤 = 𝑌 → ((((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))))
36	28, 35	rspc2ev 3601	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
37	7, 9, 18, 21, 36	syl112anc 1376	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
38		oveq1 7394	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐻𝑦) = (∅𝐻𝑦))
39	38	ineq1d 4182	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)))
40	39	neeq1d 2984	. . . . 5 ⊢ (𝑥 = ∅ → (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅))
41		eqeq1 2733	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
42	41	anbi1d 631	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
43	42	notbid 318	. . . . 5 ⊢ (𝑥 = ∅ → (¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
44	40, 43	anbi12d 632	. . . 4 ⊢ (𝑥 = ∅ → ((((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
45	44	2rexbidv 3202	. . 3 ⊢ (𝑥 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
46		oveq2 7395	. . . . . . 7 ⊢ (𝑦 = ∅ → (∅𝐻𝑦) = (∅𝐻∅))
47	46	ineq1d 4182	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (𝑧𝐻𝑤)))
48	47	neeq1d 2984	. . . . 5 ⊢ (𝑦 = ∅ → (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅))
49		eqeq1 2733	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 = 𝑤 ↔ ∅ = 𝑤))
50	49	anbi2d 630	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ ∅ = 𝑤)))
51	50	notbid 318	. . . . 5 ⊢ (𝑦 = ∅ → (¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
52	48, 51	anbi12d 632	. . . 4 ⊢ (𝑦 = ∅ → ((((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
53	52	2rexbidv 3202	. . 3 ⊢ (𝑦 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
54	45, 53	rspc2ev 3601	. 2 ⊢ ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
55	7, 7, 37, 54	syl3anc 1373	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∩ cin 3913  ∅c0 4296  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  SetCatcsetc 18037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-setc 18038

This theorem is referenced by:  cat1  18059