Theorem cat1lem 18153

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 18154. (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 18135	. . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
5		cat1lem.3	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2822	. . 3 ⊢ (𝜑 → 𝑈 = 𝐵)
7	1, 6	eleqtrd 2871	. 2 ⊢ (𝜑 → ∅ ∈ 𝐵)
8		cat1lem.6	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
9	8, 6	eleqtrd 2871	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		f0 6760	. . . . 5 ⊢ ∅:∅⟶∅
11		cat1lem.4	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
12	2, 3, 11, 1, 1	elsetchom 18138	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅))
13	10, 12	mpbiri 261	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻∅))
14		f0 6760	. . . . 5 ⊢ ∅:∅⟶𝑌
15	2, 3, 11, 1, 8	elsetchom 18138	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻𝑌) ↔ ∅:∅⟶𝑌))
16	14, 15	mpbiri 261	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻𝑌))
17		inelcm 4431	. . . 4 ⊢ ((∅ ∈ (∅𝐻∅) ∧ ∅ ∈ (∅𝐻𝑌)) → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
18	13, 16, 17	syl2anc 595	. . 3 ⊢ (𝜑 → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
19		cat1lem.7	. . . . 5 ⊢ (𝜑 → ∅ ≠ 𝑌)
20	19	neneqd 2969	. . . 4 ⊢ (𝜑 → ¬ ∅ = 𝑌)
21	20	intnand 493	. . 3 ⊢ (𝜑 → ¬ (∅ = ∅ ∧ ∅ = 𝑌))
22		oveq1 7418	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧𝐻𝑤) = (∅𝐻𝑤))
23	22	ineq2d 4181	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑤)))
24	23	neeq1d 3023	. . . . 5 ⊢ (𝑧 = ∅ → (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅))
25		eqeq2 2781	. . . . . . 7 ⊢ (𝑧 = ∅ → (∅ = 𝑧 ↔ ∅ = ∅))
26	25	anbi1d 642	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅ = 𝑧 ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑤)))
27	26	notbid 321	. . . . 5 ⊢ (𝑧 = ∅ → (¬ (∅ = 𝑧 ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑤)))
28	24, 27	anbi12d 643	. . . 4 ⊢ (𝑧 = ∅ → ((((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤))))
29		oveq2 7419	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅𝐻𝑤) = (∅𝐻𝑌))
30	29	ineq2d 4181	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅𝐻∅) ∩ (∅𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑌)))
31	30	neeq1d 3023	. . . . 5 ⊢ (𝑤 = 𝑌 → (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅))
32		eqeq2 2781	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅ = 𝑤 ↔ ∅ = 𝑌))
33	32	anbi2d 641	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅ = ∅ ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑌)))
34	33	notbid 321	. . . . 5 ⊢ (𝑤 = 𝑌 → (¬ (∅ = ∅ ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑌)))
35	31, 34	anbi12d 643	. . . 4 ⊢ (𝑤 = 𝑌 → ((((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))))
36	28, 35	rspc2ev 3603	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
37	7, 9, 18, 21, 36	syl112anc 1399	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
38		oveq1 7418	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐻𝑦) = (∅𝐻𝑦))
39	38	ineq1d 4180	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)))
40	39	neeq1d 3023	. . . . 5 ⊢ (𝑥 = ∅ → (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅))
41		eqeq1 2773	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
42	41	anbi1d 642	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
43	42	notbid 321	. . . . 5 ⊢ (𝑥 = ∅ → (¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
44	40, 43	anbi12d 643	. . . 4 ⊢ (𝑥 = ∅ → ((((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
45	44	2rexbidv 3236	. . 3 ⊢ (𝑥 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
46		oveq2 7419	. . . . . . 7 ⊢ (𝑦 = ∅ → (∅𝐻𝑦) = (∅𝐻∅))
47	46	ineq1d 4180	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (𝑧𝐻𝑤)))
48	47	neeq1d 3023	. . . . 5 ⊢ (𝑦 = ∅ → (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅))
49		eqeq1 2773	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 = 𝑤 ↔ ∅ = 𝑤))
50	49	anbi2d 641	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ ∅ = 𝑤)))
51	50	notbid 321	. . . . 5 ⊢ (𝑦 = ∅ → (¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
52	48, 51	anbi12d 643	. . . 4 ⊢ (𝑦 = ∅ → ((((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
53	52	2rexbidv 3236	. . 3 ⊢ (𝑦 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
54	45, 53	rspc2ev 3603	. 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 1396	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∩ cin 3912  ∅c0 4294  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Hom chom 17321  SetCatcsetc 18132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-setc 18133

This theorem is referenced by:  cat1  18154