Theorem cat1lem 18130

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 18131. (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 18112	. . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
5		cat1lem.3	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2816	. . 3 ⊢ (𝜑 → 𝑈 = 𝐵)
7	1, 6	eleqtrd 2865	. 2 ⊢ (𝜑 → ∅ ∈ 𝐵)
8		cat1lem.6	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
9	8, 6	eleqtrd 2865	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		f0 6746	. . . . 5 ⊢ ∅:∅⟶∅
11		cat1lem.4	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
12	2, 3, 11, 1, 1	elsetchom 18115	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅))
13	10, 12	mpbiri 260	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻∅))
14		f0 6746	. . . . 5 ⊢ ∅:∅⟶𝑌
15	2, 3, 11, 1, 8	elsetchom 18115	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻𝑌) ↔ ∅:∅⟶𝑌))
16	14, 15	mpbiri 260	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻𝑌))
17		inelcm 4420	. . . 4 ⊢ ((∅ ∈ (∅𝐻∅) ∧ ∅ ∈ (∅𝐻𝑌)) → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
18	13, 16, 17	syl2anc 593	. . 3 ⊢ (𝜑 → ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅)
19		cat1lem.7	. . . . 5 ⊢ (𝜑 → ∅ ≠ 𝑌)
20	19	neneqd 2963	. . . 4 ⊢ (𝜑 → ¬ ∅ = 𝑌)
21	20	intnand 492	. . 3 ⊢ (𝜑 → ¬ (∅ = ∅ ∧ ∅ = 𝑌))
22		oveq1 7404	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧𝐻𝑤) = (∅𝐻𝑤))
23	22	ineq2d 4173	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑤)))
24	23	neeq1d 3017	. . . . 5 ⊢ (𝑧 = ∅ → (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅))
25		eqeq2 2775	. . . . . . 7 ⊢ (𝑧 = ∅ → (∅ = 𝑧 ↔ ∅ = ∅))
26	25	anbi1d 640	. . . . . 6 ⊢ (𝑧 = ∅ → ((∅ = 𝑧 ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑤)))
27	26	notbid 320	. . . . 5 ⊢ (𝑧 = ∅ → (¬ (∅ = 𝑧 ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑤)))
28	24, 27	anbi12d 641	. . . 4 ⊢ (𝑧 = ∅ → ((((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤))))
29		oveq2 7405	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅𝐻𝑤) = (∅𝐻𝑌))
30	29	ineq2d 4173	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅𝐻∅) ∩ (∅𝐻𝑤)) = ((∅𝐻∅) ∩ (∅𝐻𝑌)))
31	30	neeq1d 3017	. . . . 5 ⊢ (𝑤 = 𝑌 → (((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅))
32		eqeq2 2775	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅ = 𝑤 ↔ ∅ = 𝑌))
33	32	anbi2d 639	. . . . . 6 ⊢ (𝑤 = 𝑌 → ((∅ = ∅ ∧ ∅ = 𝑤) ↔ (∅ = ∅ ∧ ∅ = 𝑌)))
34	33	notbid 320	. . . . 5 ⊢ (𝑤 = 𝑌 → (¬ (∅ = ∅ ∧ ∅ = 𝑤) ↔ ¬ (∅ = ∅ ∧ ∅ = 𝑌)))
35	31, 34	anbi12d 641	. . . 4 ⊢ (𝑤 = 𝑌 → ((((∅𝐻∅) ∩ (∅𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑤)) ↔ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))))
36	28, 35	rspc2ev 3595	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (((∅𝐻∅) ∩ (∅𝐻𝑌)) ≠ ∅ ∧ ¬ (∅ = ∅ ∧ ∅ = 𝑌))) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
37	7, 9, 18, 21, 36	syl112anc 1394	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
38		oveq1 7404	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐻𝑦) = (∅𝐻𝑦))
39	38	ineq1d 4172	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)))
40	39	neeq1d 3017	. . . . 5 ⊢ (𝑥 = ∅ → (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅))
41		eqeq1 2767	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
42	41	anbi1d 640	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
43	42	notbid 320	. . . . 5 ⊢ (𝑥 = ∅ → (¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)))
44	40, 43	anbi12d 641	. . . 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 7405	. . . . . . 7 ⊢ (𝑦 = ∅ → (∅𝐻𝑦) = (∅𝐻∅))
47	46	ineq1d 4172	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) = ((∅𝐻∅) ∩ (𝑧𝐻𝑤)))
48	47	neeq1d 3017	. . . . 5 ⊢ (𝑦 = ∅ → (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ↔ ((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅))
49		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 = 𝑤 ↔ ∅ = 𝑤))
50	49	anbi2d 639	. . . . . 6 ⊢ (𝑦 = ∅ → ((∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ (∅ = 𝑧 ∧ ∅ = 𝑤)))
51	50	notbid 320	. . . . 5 ⊢ (𝑦 = ∅ → (¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤) ↔ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤)))
52	48, 51	anbi12d 641	. . . 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 3595	. 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 1391	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087   ∩ cin 3904  ∅c0 4286  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  Hom chom 17298  SetCatcsetc 18109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-er 8679  df-map 8811  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-z 12570  df-dec 12690  df-uz 12841  df-fz 13514  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17247  df-hom 17311  df-cco 17312  df-setc 18110

This theorem is referenced by:  cat1  18131