Theorem nummin 33362

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 9800	. . . . . . . 8 ⊢ card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On
2		ffun 6654	. . . . . . . . 9 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → Fun card)
3	2	funfnd 6515	. . . . . . . 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 6617	. . . . . . 7 ⊢ ((card Fn dom card ∧ 𝐴 ⊆ dom card) → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
6	4, 5	mpan 687	. . . . . 6 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
7	6	necon3bid 2985	. . . . 5 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅))
8	7	biimprd 247	. . . 4 ⊢ (𝐴 ⊆ dom card → (𝐴 ≠ ∅ → (card “ 𝐴) ≠ ∅))
9	8	imdistani 569	. . 3 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅))
10		fimass 6672	. . . . . . . . . 10 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → (card “ 𝐴) ⊆ On)
11	1, 10	ax-mp 5	. . . . . . . . 9 ⊢ (card “ 𝐴) ⊆ On
12		onssmin 7705	. . . . . . . . 9 ⊢ (((card “ 𝐴) ⊆ On ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
13	11, 12	mpan 687	. . . . . . . 8 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
14		ssel 3925	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15		ssel 3925	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑦 ∈ (card “ 𝐴) → 𝑦 ∈ On))
16	14, 15	anim12d 609	. . . . . . . . . . . 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 6336	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ On) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑧))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑧))
20		epel 5527	. . . . . . . . . . 11 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
21	20	notbii 319	. . . . . . . . . 10 ⊢ (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 ∈ 𝑧)
22	19, 21	bitr4di 288	. . . . . . . . 9 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧))
23	22	rgen2 3190	. . . . . . . 8 ⊢ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)
24		r19.29r 3115	. . . . . . . 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 586	. . . . . . 7 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)))
26		r19.26 3110	. . . . . . . . 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 220	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧) → (¬ 𝑦 E 𝑧 ↔ 𝑧 ⊆ 𝑦))
28	27	biimparc 480	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ¬ 𝑦 E 𝑧)
29	28	ralimi 3082	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
30	26, 29	sylbir 234	. . . . . . . 8 ⊢ ((∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
31	30	reximi 3083	. . . . . . 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 17	. . . . . 6 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
33	32	adantl 482	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
34		breq2 5096	. . . . . . . . . 10 ⊢ (𝑧 = (card‘𝑥) → (𝑦 E 𝑧 ↔ 𝑦 E (card‘𝑥)))
35	34	notbid 317	. . . . . . . . 9 ⊢ (𝑧 = (card‘𝑥) → (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 E (card‘𝑥)))
36	35	ralbidv 3170	. . . . . . . 8 ⊢ (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
37	36	rexima 7169	. . . . . . 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 687	. . . . . 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 481	. . . . 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 231	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
41		fvex 6838	. . . . . . . 8 ⊢ (card‘𝑥) ∈ V
42	41	dfpred3 6249	. . . . . . 7 ⊢ Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
43	42	eqeq1i 2741	. . . . . 6 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅)
44		rabeq0 4331	. . . . . 6 ⊢ ({𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
45	43, 44	bitri 274	. . . . 5 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
46	45	rexbii 3093	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
47	40, 46	sylibr 233	. . 3 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
48	9, 47	syl 17	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
49		ssel2 3927	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom card)
50		cardpred 33361	. . . . . . 7 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
51	50	eqeq1d 2738	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ (card “ Pred( ≺ , 𝐴, 𝑥)) = ∅))
52		predss 6246	. . . . . . . . 9 ⊢ Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴
53		sstr 3940	. . . . . . . . 9 ⊢ ((Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ dom card) → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
54	52, 53	mpan 687	. . . . . . . 8 ⊢ (𝐴 ⊆ dom card → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
55		fnimaeq0 6617	. . . . . . . 8 ⊢ ((card Fn dom card ∧ Pred( ≺ , 𝐴, 𝑥) ⊆ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
56	4, 54, 55	sylancr 587	. . . . . . 7 ⊢ (𝐴 ⊆ dom card → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
57	56	adantr 481	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
58	51, 57	bitrd 278	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
59	49, 58	syldan 591	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
60	59	rexbidva 3169	. . 3 ⊢ (𝐴 ⊆ dom card → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
61	60	adantr 481	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
62	48, 61	mpbid 231	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2713   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3403   ⊆ wss 3898  ∅c0 4269   class class class wbr 5092   E cep 5523  dom cdm 5620   “ cima 5623  Predcpred 6237  Oncon0 6302   Fn wfn 6474  ⟶wf 6475  ‘cfv 6479   ≈ cen 8801   ≺ csdm 8803  cardccrd 9792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-card 9796

This theorem is referenced by: (None)