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Theorem nummin 32537
 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 ≠ ∅) → ∃𝑥𝐴 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.
StepHypRef Expression
1 cardf2 9371 . . . . . . . 8 card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On
2 ffun 6495 . . . . . . . . 9 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → Fun card)
32funfnd 6360 . . . . . . . 8 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → card Fn dom card)
41, 3ax-mp 5 . . . . . . 7 card Fn dom card
5 fnimaeq0 6458 . . . . . . 7 ((card Fn dom card ∧ 𝐴 ⊆ dom card) → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
64, 5mpan 689 . . . . . 6 (𝐴 ⊆ dom card → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
76necon3bid 3031 . . . . 5 (𝐴 ⊆ dom card → ((card “ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅))
87biimprd 251 . . . 4 (𝐴 ⊆ dom card → (𝐴 ≠ ∅ → (card “ 𝐴) ≠ ∅))
98imdistani 572 . . 3 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅))
10 fimass 6534 . . . . . . . . . 10 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → (card “ 𝐴) ⊆ On)
111, 10ax-mp 5 . . . . . . . . 9 (card “ 𝐴) ⊆ On
12 onssmin 7502 . . . . . . . . 9 (((card “ 𝐴) ⊆ On ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧𝑦)
1311, 12mpan 689 . . . . . . . 8 ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧𝑦)
14 ssel 3909 . . . . . . . . . . . . 13 ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15 ssel 3909 . . . . . . . . . . . . 13 ((card “ 𝐴) ⊆ On → (𝑦 ∈ (card “ 𝐴) → 𝑦 ∈ On))
1614, 15anim12d 611 . . . . . . . . . . . 12 ((card “ 𝐴) ⊆ On → ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ∈ On ∧ 𝑦 ∈ On)))
1711, 16ax-mp 5 . . . . . . . . . . 11 ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧 ∈ On ∧ 𝑦 ∈ On))
18 ontri1 6198 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦 ∈ On) → (𝑧𝑦 ↔ ¬ 𝑦𝑧))
1917, 18syl 17 . . . . . . . . . 10 ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧𝑦 ↔ ¬ 𝑦𝑧))
20 epel 5436 . . . . . . . . . . 11 (𝑦 E 𝑧𝑦𝑧)
2120notbii 323 . . . . . . . . . 10 𝑦 E 𝑧 ↔ ¬ 𝑦𝑧)
2219, 21bitr4di 292 . . . . . . . . 9 ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧))
2322rgen2 3168 . . . . . . . 8 𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)
24 r19.29r 3217 . . . . . . . 8 ((∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)))
2513, 23, 24sylancl 589 . . . . . . 7 ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)))
26 r19.26 3137 . . . . . . . . 9 (∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) ↔ (∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)))
27 bicom1 224 . . . . . . . . . . 11 ((𝑧𝑦 ↔ ¬ 𝑦 E 𝑧) → (¬ 𝑦 E 𝑧𝑧𝑦))
2827biimparc 483 . . . . . . . . . 10 ((𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ¬ 𝑦 E 𝑧)
2928ralimi 3128 . . . . . . . . 9 (∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3026, 29sylbir 238 . . . . . . . 8 ((∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3130reximi 3206 . . . . . . 7 (∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3225, 31syl 17 . . . . . 6 ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3332adantl 485 . . . . 5 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
34 breq2 5037 . . . . . . . . . 10 (𝑧 = (card‘𝑥) → (𝑦 E 𝑧𝑦 E (card‘𝑥)))
3534notbid 321 . . . . . . . . 9 (𝑧 = (card‘𝑥) → (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 E (card‘𝑥)))
3635ralbidv 3162 . . . . . . . 8 (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
3736rexima 6984 . . . . . . 7 ((card Fn dom card ∧ 𝐴 ⊆ dom card) → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
384, 37mpan 689 . . . . . 6 (𝐴 ⊆ dom card → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
3938adantr 484 . . . . 5 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
4033, 39mpbid 235 . . . 4 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
41 fvex 6665 . . . . . . . 8 (card‘𝑥) ∈ V
4241dfpred3 6131 . . . . . . 7 Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
4342eqeq1i 2803 . . . . . 6 (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅)
44 rabeq0 4294 . . . . . 6 ({𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)} = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
4543, 44bitri 278 . . . . 5 (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
4645rexbii 3210 . . . 4 (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
4740, 46sylibr 237 . . 3 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
489, 47syl 17 . 2 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅)
49 ssel2 3911 . . . . 5 ((𝐴 ⊆ dom card ∧ 𝑥𝐴) → 𝑥 ∈ dom card)
50 cardpred 32536 . . . . . . 7 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
5150eqeq1d 2800 . . . . . 6 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ (card “ Pred( ≺ , 𝐴, 𝑥)) = ∅))
52 predss 6128 . . . . . . . . 9 Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴
53 sstr 3924 . . . . . . . . 9 ((Pred( ≺ , 𝐴, 𝑥) ⊆ 𝐴𝐴 ⊆ dom card) → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
5452, 53mpan 689 . . . . . . . 8 (𝐴 ⊆ dom card → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
55 fnimaeq0 6458 . . . . . . . 8 ((card Fn dom card ∧ Pred( ≺ , 𝐴, 𝑥) ⊆ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
564, 54, 55sylancr 590 . . . . . . 7 (𝐴 ⊆ dom card → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5756adantr 484 . . . . . 6 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5851, 57bitrd 282 . . . . 5 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5949, 58syldan 594 . . . 4 ((𝐴 ⊆ dom card ∧ 𝑥𝐴) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
6059rexbidva 3255 . . 3 (𝐴 ⊆ dom card → (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
6160adantr 484 . 2 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
6248, 61mpbid 235 1 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3882  ∅c0 4245   class class class wbr 5033   E cep 5432  dom cdm 5522   “ cima 5525  Predcpred 6120  Oncon0 6164   Fn wfn 6324  ⟶wf 6325  ‘cfv 6329   ≈ cen 8504   ≺ csdm 8506  cardccrd 9363 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-card 9367 This theorem is referenced by: (None)
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