Theorem nummin 35819

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 10054	. . . . . . . 8 ⊢ card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On
2		ffun 6737	. . . . . . . . 9 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → Fun card)
3	2	funfnd 6594	. . . . . . . 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 6697	. . . . . . 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 3003	. . . . 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 6755	. . . . . . . . . 10 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → (card “ 𝐴) ⊆ On)
11	1, 10	ax-mp 5	. . . . . . . . 9 ⊢ (card “ 𝐴) ⊆ On
12		onssmin 7833	. . . . . . . . 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 3907	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15		ssel 3907	. . . . . . . . . . . . 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 6418	. . . . . . . . . . 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 5559	. . . . . . . . . . 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 3206	. . . . . . . 8 ⊢ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)
24		r19.29r 3130	. . . . . . . 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 3126	. . . . . . . . 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 3103	. . . . . . . . 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 3104	. . . . . . 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 5110	. . . . . . . . . 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 3189	. . . . . . . 8 ⊢ (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
37	36	rexima 7270	. . . . . . 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 6924	. . . . . . . 8 ⊢ (card‘𝑥) ∈ V
42	41	dfpred3 6336	. . . . . . 7 ⊢ Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
43	42	eqeq1i 2768	. . . . . 6 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅)
44		rabeq0 4341	. . . . . 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 3113	. . . 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 3908	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom card)
50		cardpred 35818	. . . . . . 7 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
51	50	eqeq1d 2765	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ (card “ Pred( ≺ , 𝐴, 𝑥)) = ∅))
52		predss 6333	. . . . . . . . 9 ⊢ Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴
53		sstr 3921	. . . . . . . . 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 6697	. . . . . . . 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 3188	. . 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 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106   E cep 5555  dom cdm 5656   “ cima 5659  Predcpred 6324  Oncon0 6383   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561   ≈ cen 9001   ≺ csdm 9003  cardccrd 10046

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-card 10050

This theorem is used by: (None)