Theorem cat1lem 17727

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 17728. (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 17709	. . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
5		cat1lem.3	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2797	. . 3 ⊢ (𝜑 → 𝑈 = 𝐵)
7	1, 6	eleqtrd 2841	. 2 ⊢ (𝜑 → ∅ ∈ 𝐵)
8		cat1lem.6	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
9	8, 6	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		f0 6639	. . . . 5 ⊢ ∅:∅⟶∅
11		cat1lem.4	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
12	2, 3, 11, 1, 1	elsetchom 17712	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅))
13	10, 12	mpbiri 257	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻∅))
14		f0 6639	. . . . 5 ⊢ ∅:∅⟶𝑌
15	2, 3, 11, 1, 8	elsetchom 17712	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻𝑌) ↔ ∅:∅⟶𝑌))
16	14, 15	mpbiri 257	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻𝑌))
17		inelcm 4395	. . . 4 ⊢ ((∅ ∈ (∅𝐻∅) ∧ ∅ ∈ (∅𝐻𝑌)) → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
18	13, 16, 17	syl2anc 583	. . 3 ⊢ (𝜑 → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
19		cat1lem.7	. . . . 5 ⊢ (𝜑 → ∅ ≠ 𝑌)
20	19	neneqd 2947	. . . 4 ⊢ (𝜑 → ¬ ∅ = 𝑌)
21	20	intnand 488	. . 3 ⊢ (𝜑 → ¬ (∅ = ∅ ∧ ∅ = 𝑌))
22		oveq1 7262	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧𝐻𝑤) = (∅𝐻𝑤))
23	22	ineq2d 4143	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑤)))
24	23	neeq1d 3002	. . . . 5 ⊢ (𝑧 = ∅ → (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅))
25		eqeq2 2750	. . . . . . 7 ⊢ (𝑧 = ∅ → (∅ = 𝑧 ↔ ∅ = ∅))
26	25	anbi1d 629	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅ = 𝑧 ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑤)))
27	26	notbid 317	. . . . 5 ⊢ (𝑧 = ∅ → (¬ (∅ = 𝑧 ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑤)))
28	24, 27	anbi12d 630	. . . 4 ⊢ (𝑧 = ∅ → ((((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤))))
29		oveq2 7263	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅𝐻𝑤) = (∅𝐻𝑌))
30	29	ineq2d 4143	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅𝐻∅) ∩ (∅𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑌)))
31	30	neeq1d 3002	. . . . 5 ⊢ (𝑤 = 𝑌 → (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅))
32		eqeq2 2750	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅ = 𝑤 ↔ ∅ = 𝑌))
33	32	anbi2d 628	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅ = ∅ ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑌)))
34	33	notbid 317	. . . . 5 ⊢ (𝑤 = 𝑌 → (¬ (∅ = ∅ ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑌)))
35	31, 34	anbi12d 630	. . . 4 ⊢ (𝑤 = 𝑌 → ((((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))))
36	28, 35	rspc2ev 3564	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
37	7, 9, 18, 21, 36	syl112anc 1372	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
38		oveq1 7262	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐻𝑦) = (∅𝐻𝑦))
39	38	ineq1d 4142	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)))
40	39	neeq1d 3002	. . . . 5 ⊢ (𝑥 = ∅ → (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅))
41		eqeq1 2742	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
42	41	anbi1d 629	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
43	42	notbid 317	. . . . 5 ⊢ (𝑥 = ∅ → (¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
44	40, 43	anbi12d 630	. . . 4 ⊢ (𝑥 = ∅ → ((((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
45	44	2rexbidv 3228	. . 3 ⊢ (𝑥 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
46		oveq2 7263	. . . . . . 7 ⊢ (𝑦 = ∅ → (∅𝐻𝑦) = (∅𝐻∅))
47	46	ineq1d 4142	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (𝑧𝐻𝑤)))
48	47	neeq1d 3002	. . . . 5 ⊢ (𝑦 = ∅ → (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅))
49		eqeq1 2742	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 = 𝑤 ↔ ∅ = 𝑤))
50	49	anbi2d 628	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ ∅ = 𝑤)))
51	50	notbid 317	. . . . 5 ⊢ (𝑦 = ∅ → (¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
52	48, 51	anbi12d 630	. . . 4 ⊢ (𝑦 = ∅ → ((((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
53	52	2rexbidv 3228	. . 3 ⊢ (𝑦 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
54	45, 53	rspc2ev 3564	. 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 1369	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 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  SetCatcsetc 17706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-slot 16811  df-ndx 16823  df-base 16841  df-hom 16912  df-cco 16913  df-setc 17707

This theorem is referenced by:  cat1  17728