Theorem nummin 35509

Description: Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)

Assertion

Ref	Expression
nummin	⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem nummin

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardf2 9948	. . . . . . . 8 ⊢ card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On
2		ffun 6715	. . . . . . . . 9 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → Fun card)
3	2	funfnd 6574	. . . . . . . 8 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → card Fn dom card)
4	1, 3	ax-mp 5	. . . . . . 7 ⊢ card Fn dom card
5		fnimaeq0 6675	. . . . . . 7 ⊢ ((card Fn dom card ∧ 𝐴 ⊆ dom card) → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
6	4, 5	mpan 703	. . . . . 6 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
7	6	necon3bid 3005	. . . . 5 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅))
8	7	biimprd 251	. . . 4 ⊢ (𝐴 ⊆ dom card → (𝐴 ≠ ∅ → (card “ 𝐴) ≠ ∅))
9	8	imdistani 579	. . 3 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅))
10		fimass 6733	. . . . . . . . . 10 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → (card “ 𝐴) ⊆ On)
11	1, 10	ax-mp 5	. . . . . . . . 9 ⊢ (card “ 𝐴) ⊆ On
12		onssmin 7800	. . . . . . . . 9 ⊢ (((card “ 𝐴) ⊆ On ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
13	11, 12	mpan 703	. . . . . . . 8 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
14		ssel 3934	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15		ssel 3934	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑦 ∈ (card “ 𝐴) → 𝑦 ∈ On))
16	14, 15	anim12d 621	. . . . . . . . . . . 12 ⊢ ((card “ 𝐴) ⊆ On → ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ∈ On ∧ 𝑦 ∈ On)))
17	11, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ∈ On ∧ 𝑦 ∈ On))
18		ontri1 6402	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ On) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑧))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑧))
20		epel 5569	. . . . . . . . . . 11 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
21	20	notbii 323	. . . . . . . . . 10 ⊢ (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 ∈ 𝑧)
22	19, 21	bitr4di 292	. . . . . . . . 9 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧))
23	22	rgen2 3208	. . . . . . . 8 ⊢ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)
24		r19.29r 3132	. . . . . . . 8 ⊢ ((∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)))
25	13, 23, 24	sylancl 598	. . . . . . 7 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)))
26		r19.26 3128	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) ↔ (∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)))
27		bicom1 224	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧) → (¬ 𝑦 E 𝑧 ↔ 𝑧 ⊆ 𝑦))
28	27	biimparc 485	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ¬ 𝑦 E 𝑧)
29	28	ralimi 3105	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
30	26, 29	sylbir 238	. . . . . . . 8 ⊢ ((∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
31	30	reximi 3106	. . . . . . 7 ⊢ (∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
32	25, 31	syl 18	. . . . . 6 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
33	32	adantl 487	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
34		breq2 5118	. . . . . . . . . 10 ⊢ (𝑧 = (card‘𝑥) → (𝑦 E 𝑧 ↔ 𝑦 E (card‘𝑥)))
35	34	notbid 321	. . . . . . . . 9 ⊢ (𝑧 = (card‘𝑥) → (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 E (card‘𝑥)))
36	35	ralbidv 3191	. . . . . . . 8 ⊢ (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
37	36	rexima 7243	. . . . . . 7 ⊢ ((card Fn dom card ∧ 𝐴 ⊆ dom card) → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
38	4, 37	mpan 703	. . . . . 6 ⊢ (𝐴 ⊆ dom card → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
39	38	adantr 486	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
40	33, 39	mpbid 235	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
41		fvex 6901	. . . . . . . 8 ⊢ (card‘𝑥) ∈ V
42	41	dfpred3 6320	. . . . . . 7 ⊢ Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
43	42	eqeq1i 2771	. . . . . 6 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅)
44		rabeq0 4348	. . . . . 6 ⊢ ({𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
45	43, 44	bitri 278	. . . . 5 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
46	45	rexbii 3115	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
47	40, 46	sylibr 237	. . 3 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
48	9, 47	syl 18	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
49		ssel2 3935	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom card)
50		cardpred 35508	. . . . . . 7 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
51	50	eqeq1d 2768	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ (card “ Pred( ≺ , 𝐴, 𝑥)) = ∅))
52		predss 6317	. . . . . . . . 9 ⊢ Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴
53		sstr 3948	. . . . . . . . 9 ⊢ ((Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ dom card) → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
54	52, 53	mpan 703	. . . . . . . 8 ⊢ (𝐴 ⊆ dom card → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
55		fnimaeq0 6675	. . . . . . . 8 ⊢ ((card Fn dom card ∧ Pred( ≺ , 𝐴, 𝑥) ⊆ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
56	4, 54, 55	sylancr 599	. . . . . . 7 ⊢ (𝐴 ⊆ dom card → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
57	56	adantr 486	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
58	51, 57	bitrd 282	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
59	49, 58	syldan 603	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
60	59	rexbidva 3190	. . 3 ⊢ (𝐴 ⊆ dom card → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
61	60	adantr 486	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
62	48, 61	mpbid 235	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  {crab 3419   ⊆ wss 3908  ∅c0 4289   class class class wbr 5114   E cep 5565  dom cdm 5666   “ cima 5669  Predcpred 6308  Oncon0 6367   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543   ≈ cen 8949   ≺ csdm 8951  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-card 9944

This theorem is used by: (None)