Theorem nummin 35268

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 9867	. . . . . . . 8 ⊢ card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On
2		ffun 6673	. . . . . . . . 9 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → Fun card)
3	2	funfnd 6531	. . . . . . . 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 6633	. . . . . . 7 ⊢ ((card Fn dom card ∧ 𝐴 ⊆ dom card) → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
6	4, 5	mpan 691	. . . . . 6 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
7	6	necon3bid 2977	. . . . 5 ⊢ (𝐴 ⊆ dom card → ((card “ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅))
8	7	biimprd 248	. . . 4 ⊢ (𝐴 ⊆ dom card → (𝐴 ≠ ∅ → (card “ 𝐴) ≠ ∅))
9	8	imdistani 568	. . 3 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅))
10		fimass 6690	. . . . . . . . . 10 ⊢ (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑧}⟶On → (card “ 𝐴) ⊆ On)
11	1, 10	ax-mp 5	. . . . . . . . 9 ⊢ (card “ 𝐴) ⊆ On
12		onssmin 7747	. . . . . . . . 9 ⊢ (((card “ 𝐴) ⊆ On ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
13	11, 12	mpan 691	. . . . . . . 8 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦)
14		ssel 3929	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15		ssel 3929	. . . . . . . . . . . . 13 ⊢ ((card “ 𝐴) ⊆ On → (𝑦 ∈ (card “ 𝐴) → 𝑦 ∈ On))
16	14, 15	anim12d 610	. . . . . . . . . . . 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 6359	. . . . . . . . . . 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 5535	. . . . . . . . . . 11 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
21	20	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 ∈ 𝑧)
22	19, 21	bitr4di 289	. . . . . . . . 9 ⊢ ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧))
23	22	rgen2 3178	. . . . . . . 8 ⊢ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)
24		r19.29r 3102	. . . . . . . 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 587	. . . . . . 7 ⊢ ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)))
26		r19.26 3098	. . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧) → (¬ 𝑦 E 𝑧 ↔ 𝑧 ⊆ 𝑦))
28	27	biimparc 479	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ¬ 𝑦 E 𝑧)
29	28	ralimi 3075	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ∧ (𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
30	26, 29	sylbir 235	. . . . . . . 8 ⊢ ((∀𝑦 ∈ (card “ 𝐴)𝑧 ⊆ 𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧 ⊆ 𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
31	30	reximi 3076	. . . . . . 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 481	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
34		breq2 5104	. . . . . . . . . 10 ⊢ (𝑧 = (card‘𝑥) → (𝑦 E 𝑧 ↔ 𝑦 E (card‘𝑥)))
35	34	notbid 318	. . . . . . . . 9 ⊢ (𝑧 = (card‘𝑥) → (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 E (card‘𝑥)))
36	35	ralbidv 3161	. . . . . . . 8 ⊢ (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
37	36	rexima 7194	. . . . . . 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 691	. . . . . 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 480	. . . . 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 232	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
41		fvex 6855	. . . . . . . 8 ⊢ (card‘𝑥) ∈ V
42	41	dfpred3 6278	. . . . . . 7 ⊢ Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
43	42	eqeq1i 2742	. . . . . 6 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅)
44		rabeq0 4342	. . . . . 6 ⊢ ({𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
45	43, 44	bitri 275	. . . . 5 ⊢ (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
46	45	rexbii 3085	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
47	40, 46	sylibr 234	. . 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 3930	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom card)
50		cardpred 35267	. . . . . . 7 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
51	50	eqeq1d 2739	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ (card “ Pred( ≺ , 𝐴, 𝑥)) = ∅))
52		predss 6275	. . . . . . . . 9 ⊢ Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴
53		sstr 3944	. . . . . . . . 9 ⊢ ((Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ dom card) → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
54	52, 53	mpan 691	. . . . . . . 8 ⊢ (𝐴 ⊆ dom card → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
55		fnimaeq0 6633	. . . . . . . 8 ⊢ ((card Fn dom card ∧ Pred( ≺ , 𝐴, 𝑥) ⊆ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
56	4, 54, 55	sylancr 588	. . . . . . 7 ⊢ (𝐴 ⊆ dom card → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
57	56	adantr 480	. . . . . 6 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
58	51, 57	bitrd 279	. . . . 5 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
59	49, 58	syldan 592	. . . 4 ⊢ ((𝐴 ⊆ dom card ∧ 𝑥 ∈ 𝐴) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
60	59	rexbidva 3160	. . 3 ⊢ (𝐴 ⊆ dom card → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
61	60	adantr 480	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
62	48, 61	mpbid 232	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287   class class class wbr 5100   E cep 5531  dom cdm 5632   “ cima 5635  Predcpred 6266  Oncon0 6325   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500   ≈ cen 8892   ≺ csdm 8894  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-card 9863

This theorem is referenced by: (None)