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Theorem nummin 34094
Description: Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)
Assertion
Ref Expression
nummin ((𝐴 βŠ† dom card ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ 𝐴 Pred( β‰Ί , 𝐴, π‘₯) = βˆ…)
Distinct variable group:   π‘₯,𝐴

Proof of Theorem nummin
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 9938 . . . . . . . 8 card:{𝑧 ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧}⟢On
2 ffun 6721 . . . . . . . . 9 (card:{𝑧 ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧}⟢On β†’ Fun card)
32funfnd 6580 . . . . . . . 8 (card:{𝑧 ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧}⟢On β†’ card Fn dom card)
41, 3ax-mp 5 . . . . . . 7 card Fn dom card
5 fnimaeq0 6684 . . . . . . 7 ((card Fn dom card ∧ 𝐴 βŠ† dom card) β†’ ((card β€œ 𝐴) = βˆ… ↔ 𝐴 = βˆ…))
64, 5mpan 689 . . . . . 6 (𝐴 βŠ† dom card β†’ ((card β€œ 𝐴) = βˆ… ↔ 𝐴 = βˆ…))
76necon3bid 2986 . . . . 5 (𝐴 βŠ† dom card β†’ ((card β€œ 𝐴) β‰  βˆ… ↔ 𝐴 β‰  βˆ…))
87biimprd 247 . . . 4 (𝐴 βŠ† dom card β†’ (𝐴 β‰  βˆ… β†’ (card β€œ 𝐴) β‰  βˆ…))
98imdistani 570 . . 3 ((𝐴 βŠ† dom card ∧ 𝐴 β‰  βˆ…) β†’ (𝐴 βŠ† dom card ∧ (card β€œ 𝐴) β‰  βˆ…))
10 fimass 6739 . . . . . . . . . 10 (card:{𝑧 ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧}⟢On β†’ (card β€œ 𝐴) βŠ† On)
111, 10ax-mp 5 . . . . . . . . 9 (card β€œ 𝐴) βŠ† On
12 onssmin 7780 . . . . . . . . 9 (((card β€œ 𝐴) βŠ† On ∧ (card β€œ 𝐴) β‰  βˆ…) β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦)
1311, 12mpan 689 . . . . . . . 8 ((card β€œ 𝐴) β‰  βˆ… β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦)
14 ssel 3976 . . . . . . . . . . . . 13 ((card β€œ 𝐴) βŠ† On β†’ (𝑧 ∈ (card β€œ 𝐴) β†’ 𝑧 ∈ On))
15 ssel 3976 . . . . . . . . . . . . 13 ((card β€œ 𝐴) βŠ† On β†’ (𝑦 ∈ (card β€œ 𝐴) β†’ 𝑦 ∈ On))
1614, 15anim12d 610 . . . . . . . . . . . 12 ((card β€œ 𝐴) βŠ† On β†’ ((𝑧 ∈ (card β€œ 𝐴) ∧ 𝑦 ∈ (card β€œ 𝐴)) β†’ (𝑧 ∈ On ∧ 𝑦 ∈ On)))
1711, 16ax-mp 5 . . . . . . . . . . 11 ((𝑧 ∈ (card β€œ 𝐴) ∧ 𝑦 ∈ (card β€œ 𝐴)) β†’ (𝑧 ∈ On ∧ 𝑦 ∈ On))
18 ontri1 6399 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦 ∈ On) β†’ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ 𝑧))
1917, 18syl 17 . . . . . . . . . 10 ((𝑧 ∈ (card β€œ 𝐴) ∧ 𝑦 ∈ (card β€œ 𝐴)) β†’ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ 𝑧))
20 epel 5584 . . . . . . . . . . 11 (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
2120notbii 320 . . . . . . . . . 10 (Β¬ 𝑦 E 𝑧 ↔ Β¬ 𝑦 ∈ 𝑧)
2219, 21bitr4di 289 . . . . . . . . 9 ((𝑧 ∈ (card β€œ 𝐴) ∧ 𝑦 ∈ (card β€œ 𝐴)) β†’ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧))
2322rgen2 3198 . . . . . . . 8 βˆ€π‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)
24 r19.29r 3117 . . . . . . . 8 ((βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)(βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)))
2513, 23, 24sylancl 587 . . . . . . 7 ((card β€œ 𝐴) β‰  βˆ… β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)(βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)))
26 r19.26 3112 . . . . . . . . 9 (βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ∧ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) ↔ (βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)))
27 bicom1 220 . . . . . . . . . . 11 ((𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧) β†’ (Β¬ 𝑦 E 𝑧 ↔ 𝑧 βŠ† 𝑦))
2827biimparc 481 . . . . . . . . . 10 ((𝑧 βŠ† 𝑦 ∧ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) β†’ Β¬ 𝑦 E 𝑧)
2928ralimi 3084 . . . . . . . . 9 (βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ∧ (𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) β†’ βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧)
3026, 29sylbir 234 . . . . . . . 8 ((βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) β†’ βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧)
3130reximi 3085 . . . . . . 7 (βˆƒπ‘§ ∈ (card β€œ 𝐴)(βˆ€π‘¦ ∈ (card β€œ 𝐴)𝑧 βŠ† 𝑦 ∧ βˆ€π‘¦ ∈ (card β€œ 𝐴)(𝑧 βŠ† 𝑦 ↔ Β¬ 𝑦 E 𝑧)) β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧)
3225, 31syl 17 . . . . . 6 ((card β€œ 𝐴) β‰  βˆ… β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧)
3332adantl 483 . . . . 5 ((𝐴 βŠ† dom card ∧ (card β€œ 𝐴) β‰  βˆ…) β†’ βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧)
34 breq2 5153 . . . . . . . . . 10 (𝑧 = (cardβ€˜π‘₯) β†’ (𝑦 E 𝑧 ↔ 𝑦 E (cardβ€˜π‘₯)))
3534notbid 318 . . . . . . . . 9 (𝑧 = (cardβ€˜π‘₯) β†’ (Β¬ 𝑦 E 𝑧 ↔ Β¬ 𝑦 E (cardβ€˜π‘₯)))
3635ralbidv 3178 . . . . . . . 8 (𝑧 = (cardβ€˜π‘₯) β†’ (βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧 ↔ βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯)))
3736rexima 7239 . . . . . . 7 ((card Fn dom card ∧ 𝐴 βŠ† dom card) β†’ (βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧 ↔ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯)))
384, 37mpan 689 . . . . . 6 (𝐴 βŠ† dom card β†’ (βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧 ↔ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯)))
3938adantr 482 . . . . 5 ((𝐴 βŠ† dom card ∧ (card β€œ 𝐴) β‰  βˆ…) β†’ (βˆƒπ‘§ ∈ (card β€œ 𝐴)βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E 𝑧 ↔ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯)))
4033, 39mpbid 231 . . . 4 ((𝐴 βŠ† dom card ∧ (card β€œ 𝐴) β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯))
41 fvex 6905 . . . . . . . 8 (cardβ€˜π‘₯) ∈ V
4241dfpred3 6312 . . . . . . 7 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = {𝑦 ∈ (card β€œ 𝐴) ∣ 𝑦 E (cardβ€˜π‘₯)}
4342eqeq1i 2738 . . . . . 6 (Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ {𝑦 ∈ (card β€œ 𝐴) ∣ 𝑦 E (cardβ€˜π‘₯)} = βˆ…)
44 rabeq0 4385 . . . . . 6 ({𝑦 ∈ (card β€œ 𝐴) ∣ 𝑦 E (cardβ€˜π‘₯)} = βˆ… ↔ βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯))
4543, 44bitri 275 . . . . 5 (Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯))
4645rexbii 3095 . . . 4 (βˆƒπ‘₯ ∈ 𝐴 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ (card β€œ 𝐴) Β¬ 𝑦 E (cardβ€˜π‘₯))
4740, 46sylibr 233 . . 3 ((𝐴 βŠ† dom card ∧ (card β€œ 𝐴) β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ 𝐴 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ…)
489, 47syl 17 . 2 ((𝐴 βŠ† dom card ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ 𝐴 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ…)
49 ssel2 3978 . . . . 5 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ dom card)
50 cardpred 34093 . . . . . . 7 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ dom card) β†’ Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = (card β€œ Pred( β‰Ί , 𝐴, π‘₯)))
5150eqeq1d 2735 . . . . . 6 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ dom card) β†’ (Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ (card β€œ Pred( β‰Ί , 𝐴, π‘₯)) = βˆ…))
52 predss 6309 . . . . . . . . 9 Pred( β‰Ί , 𝐴, π‘₯) βŠ† 𝐴
53 sstr 3991 . . . . . . . . 9 ((Pred( β‰Ί , 𝐴, π‘₯) βŠ† 𝐴 ∧ 𝐴 βŠ† dom card) β†’ Pred( β‰Ί , 𝐴, π‘₯) βŠ† dom card)
5452, 53mpan 689 . . . . . . . 8 (𝐴 βŠ† dom card β†’ Pred( β‰Ί , 𝐴, π‘₯) βŠ† dom card)
55 fnimaeq0 6684 . . . . . . . 8 ((card Fn dom card ∧ Pred( β‰Ί , 𝐴, π‘₯) βŠ† dom card) β†’ ((card β€œ Pred( β‰Ί , 𝐴, π‘₯)) = βˆ… ↔ Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
564, 54, 55sylancr 588 . . . . . . 7 (𝐴 βŠ† dom card β†’ ((card β€œ Pred( β‰Ί , 𝐴, π‘₯)) = βˆ… ↔ Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
5756adantr 482 . . . . . 6 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ dom card) β†’ ((card β€œ Pred( β‰Ί , 𝐴, π‘₯)) = βˆ… ↔ Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
5851, 57bitrd 279 . . . . 5 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ dom card) β†’ (Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
5949, 58syldan 592 . . . 4 ((𝐴 βŠ† dom card ∧ π‘₯ ∈ 𝐴) β†’ (Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
6059rexbidva 3177 . . 3 (𝐴 βŠ† dom card β†’ (βˆƒπ‘₯ ∈ 𝐴 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐴 Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
6160adantr 482 . 2 ((𝐴 βŠ† dom card ∧ 𝐴 β‰  βˆ…) β†’ (βˆƒπ‘₯ ∈ 𝐴 Pred( E , (card β€œ 𝐴), (cardβ€˜π‘₯)) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐴 Pred( β‰Ί , 𝐴, π‘₯) = βˆ…))
6248, 61mpbid 231 1 ((𝐴 βŠ† dom card ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ 𝐴 Pred( β‰Ί , 𝐴, π‘₯) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149   E cep 5580  dom cdm 5677   β€œ cima 5680  Predcpred 6300  Oncon0 6365   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544   β‰ˆ cen 8936   β‰Ί csdm 8938  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9934
This theorem is referenced by: (None)
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