Theorem cat1lem 18038

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 18039. (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 18020	. . . 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 6723	. . . . 5 ⊢ ∅:∅⟶∅
11		cat1lem.4	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
12	2, 3, 11, 1, 1	elsetchom 18023	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅))
13	10, 12	mpbiri 258	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻∅))
14		f0 6723	. . . . 5 ⊢ ∅:∅⟶𝑌
15	2, 3, 11, 1, 8	elsetchom 18023	. . . . 5 ⊢ (𝜑 → (∅ ∈ (∅𝐻𝑌) ↔ ∅:∅⟶𝑌))
16	14, 15	mpbiri 258	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅𝐻𝑌))
17		inelcm 4424	. . . 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 7376	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧𝐻𝑤) = (∅𝐻𝑤))
23	22	ineq2d 4179	. . . . . 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 7377	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (∅𝐻𝑤) = (∅𝐻𝑌))
30	29	ineq2d 4179	. . . . . 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 3598	. . 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 7376	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐻𝑦) = (∅𝐻𝑦))
39	38	ineq1d 4178	. . . . . 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 3200	. . 3 ⊢ (𝑥 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤))))
46		oveq2 7377	. . . . . . 7 ⊢ (𝑦 = ∅ → (∅𝐻𝑦) = (∅𝐻∅))
47	46	ineq1d 4178	. . . . . 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 3200	. . 3 ⊢ (𝑦 = ∅ → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((∅𝐻∅) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (∅ = 𝑧 ∧ ∅ = 𝑤))))
54	45, 53	rspc2ev 3598	. 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 3910  ∅c0 4292  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  SetCatcsetc 18017

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-setc 18018

This theorem is referenced by:  cat1  18039