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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 ≠ ∅) → ∃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 9800 . . . . . . . 8 card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On
2 ffun 6654 . . . . . . . . 9 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → Fun card)
32funfnd 6515 . . . . . . . 8 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → card Fn dom card)
41, 3ax-mp 5 . . . . . . 7 card Fn dom card
5 fnimaeq0 6617 . . . . . . 7 ((card Fn dom card ∧ 𝐴 ⊆ dom card) → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
64, 5mpan 687 . . . . . 6 (𝐴 ⊆ dom card → ((card “ 𝐴) = ∅ ↔ 𝐴 = ∅))
76necon3bid 2985 . . . . 5 (𝐴 ⊆ dom card → ((card “ 𝐴) ≠ ∅ ↔ 𝐴 ≠ ∅))
87biimprd 247 . . . 4 (𝐴 ⊆ dom card → (𝐴 ≠ ∅ → (card “ 𝐴) ≠ ∅))
98imdistani 569 . . 3 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅))
10 fimass 6672 . . . . . . . . . 10 (card:{𝑧 ∣ ∃𝑦 ∈ On 𝑦𝑧}⟶On → (card “ 𝐴) ⊆ On)
111, 10ax-mp 5 . . . . . . . . 9 (card “ 𝐴) ⊆ On
12 onssmin 7705 . . . . . . . . 9 (((card “ 𝐴) ⊆ On ∧ (card “ 𝐴) ≠ ∅) → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧𝑦)
1311, 12mpan 687 . . . . . . . 8 ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)𝑧𝑦)
14 ssel 3925 . . . . . . . . . . . . 13 ((card “ 𝐴) ⊆ On → (𝑧 ∈ (card “ 𝐴) → 𝑧 ∈ On))
15 ssel 3925 . . . . . . . . . . . . 13 ((card “ 𝐴) ⊆ On → (𝑦 ∈ (card “ 𝐴) → 𝑦 ∈ On))
1614, 15anim12d 609 . . . . . . . . . . . 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 6336 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦 ∈ On) → (𝑧𝑦 ↔ ¬ 𝑦𝑧))
1917, 18syl 17 . . . . . . . . . 10 ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧𝑦 ↔ ¬ 𝑦𝑧))
20 epel 5527 . . . . . . . . . . 11 (𝑦 E 𝑧𝑦𝑧)
2120notbii 319 . . . . . . . . . 10 𝑦 E 𝑧 ↔ ¬ 𝑦𝑧)
2219, 21bitr4di 288 . . . . . . . . 9 ((𝑧 ∈ (card “ 𝐴) ∧ 𝑦 ∈ (card “ 𝐴)) → (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧))
2322rgen2 3190 . . . . . . . 8 𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)
24 r19.29r 3115 . . . . . . . 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 586 . . . . . . 7 ((card “ 𝐴) ≠ ∅ → ∃𝑧 ∈ (card “ 𝐴)(∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)))
26 r19.26 3110 . . . . . . . . 9 (∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) ↔ (∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)))
27 bicom1 220 . . . . . . . . . . 11 ((𝑧𝑦 ↔ ¬ 𝑦 E 𝑧) → (¬ 𝑦 E 𝑧𝑧𝑦))
2827biimparc 480 . . . . . . . . . 10 ((𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ¬ 𝑦 E 𝑧)
2928ralimi 3082 . . . . . . . . 9 (∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ∧ (𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3026, 29sylbir 234 . . . . . . . 8 ((∀𝑦 ∈ (card “ 𝐴)𝑧𝑦 ∧ ∀𝑦 ∈ (card “ 𝐴)(𝑧𝑦 ↔ ¬ 𝑦 E 𝑧)) → ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧)
3130reximi 3083 . . . . . . 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 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 𝑧𝑦 E (card‘𝑥)))
3534notbid 317 . . . . . . . . 9 (𝑧 = (card‘𝑥) → (¬ 𝑦 E 𝑧 ↔ ¬ 𝑦 E (card‘𝑥)))
3635ralbidv 3170 . . . . . . . 8 (𝑧 = (card‘𝑥) → (∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
3736rexima 7169 . . . . . . 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 687 . . . . . 6 (𝐴 ⊆ dom card → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
3938adantr 481 . . . . 5 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → (∃𝑧 ∈ (card “ 𝐴)∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E 𝑧 ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥)))
4033, 39mpbid 231 . . . 4 ((𝐴 ⊆ dom card ∧ (card “ 𝐴) ≠ ∅) → ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
41 fvex 6838 . . . . . . . 8 (card‘𝑥) ∈ V
4241dfpred3 6249 . . . . . . 7 Pred( E , (card “ 𝐴), (card‘𝑥)) = {𝑦 ∈ (card “ 𝐴) ∣ 𝑦 E (card‘𝑥)}
4342eqeq1i 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))
4543, 44bitri 274 . . . . 5 (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∀𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
4645rexbii 3093 . . . 4 (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴𝑦 ∈ (card “ 𝐴) ¬ 𝑦 E (card‘𝑥))
4740, 46sylibr 233 . . 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 3927 . . . . 5 ((𝐴 ⊆ dom card ∧ 𝑥𝐴) → 𝑥 ∈ dom card)
50 cardpred 33361 . . . . . . 7 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝑥)) = (card “ Pred( ≺ , 𝐴, 𝑥)))
5150eqeq1d 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) ⊆ 𝐴𝐴 ⊆ dom card) → Pred( ≺ , 𝐴, 𝑥) ⊆ dom card)
5452, 53mpan 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) = ∅))
564, 54, 55sylancr 587 . . . . . . 7 (𝐴 ⊆ dom card → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5756adantr 481 . . . . . 6 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → ((card “ Pred( ≺ , 𝐴, 𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5851, 57bitrd 278 . . . . 5 ((𝐴 ⊆ dom card ∧ 𝑥 ∈ dom card) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
5949, 58syldan 591 . . . 4 ((𝐴 ⊆ dom card ∧ 𝑥𝐴) → (Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ Pred( ≺ , 𝐴, 𝑥) = ∅))
6059rexbidva 3169 . . 3 (𝐴 ⊆ dom card → (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
6160adantr 481 . 2 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → (∃𝑥𝐴 Pred( E , (card “ 𝐴), (card‘𝑥)) = ∅ ↔ ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅))
6248, 61mpbid 231 1 ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
Colors of variables: wff setvar class
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)
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