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Theorem zorn2lem4 9600
Description: Lemma for zorn2 9607. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
zorn2lem.3 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
zorn2lem.4 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
zorn2lem.5 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
Assertion
Ref Expression
zorn2lem4 ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝐷,𝑓,𝑢,𝑣   𝑓,𝐹,𝑔,𝑢,𝑣,𝑥,𝑧   𝑅,𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶
Allowed substitution hints:   𝐶(𝑥,𝑧,𝑤,𝑢,𝑓,𝑔)   𝐷(𝑥,𝑧,𝑤,𝑔)   𝐹(𝑤)

Proof of Theorem zorn2lem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm3.24 391 . 2 ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2 df-ne 2975 . . . . 5 (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
32ralbii 3164 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4 df-ral 3097 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5 ralnex 3176 . . . 4 (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
63, 4, 53bitr3i 292 . . 3 (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7 weso 5296 . . . . . . . . 9 (𝑤 We 𝐴𝑤 Or 𝐴)
87adantr 468 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9 vex 3390 . . . . . . . 8 𝑤 ∈ V
10 soex 7333 . . . . . . . 8 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
118, 9, 10sylancl 576 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12 zorn2lem.3 . . . . . . . . . . 11 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
1312tfr1 7723 . . . . . . . . . 10 𝐹 Fn On
14 fvelrnb 6458 . . . . . . . . . 10 (𝐹 Fn On → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹𝑥) = 𝑦))
1513, 14ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹𝑥) = 𝑦)
16 nfv 2005 . . . . . . . . . . 11 𝑥 𝑤 We 𝐴
17 nfa1 2194 . . . . . . . . . . 11 𝑥𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
1816, 17nfan 1990 . . . . . . . . . 10 𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19 nfv 2005 . . . . . . . . . 10 𝑥 𝑦𝐴
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
21 ssrab2 3878 . . . . . . . . . . . . . . . . . 18 {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧} ⊆ 𝐴
2220, 21eqsstri 3826 . . . . . . . . . . . . . . . . 17 𝐷𝐴
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
2412, 23, 20zorn2lem1 9597 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
2522, 24sseldi 3790 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐴)
26 eleq1 2869 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2725, 26syl5ibcom 236 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2827exp32 409 . . . . . . . . . . . . . 14 (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑦𝐴))))
2928com12 32 . . . . . . . . . . . . 13 (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3029a2d 29 . . . . . . . . . . . 12 (𝑤 We 𝐴 → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3130spsd 2221 . . . . . . . . . . 11 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3231imp 395 . . . . . . . . . 10 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
3318, 19, 32rexlimd 3210 . . . . . . . . 9 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹𝑥) = 𝑦𝑦𝐴))
3415, 33syl5bi 233 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
3534ssrdv 3798 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹𝐴)
3611, 35ssexd 4994 . . . . . 6 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
3736ex 399 . . . . 5 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3837adantl 469 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3912, 23, 20zorn2lem3 9599 . . . . . . . . . . . . . 14 ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
4039exp45 427 . . . . . . . . . . . . 13 (𝑅 Po 𝐴 → (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))))
4140com23 86 . . . . . . . . . . . 12 (𝑅 Po 𝐴 → (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))))
4241imp 395 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑤 We 𝐴) → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
4342a2d 29 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
4443imp4a 411 . . . . . . . . 9 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4544alrimdv 2019 . . . . . . . 8 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4645alimdv 2007 . . . . . . 7 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
47 r2al 3123 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
4846, 47syl6ibr 243 . . . . . 6 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)))
49 ssid 3814 . . . . . . . 8 On ⊆ On
5013tz7.48lem 7766 . . . . . . . 8 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
5149, 50mpan 673 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun (𝐹 ↾ On))
52 fnrel 6194 . . . . . . . . . . 11 (𝐹 Fn On → Rel 𝐹)
5313, 52ax-mp 5 . . . . . . . . . 10 Rel 𝐹
54 fndm 6195 . . . . . . . . . . . 12 (𝐹 Fn On → dom 𝐹 = On)
5513, 54ax-mp 5 . . . . . . . . . . 11 dom 𝐹 = On
5655eqimssi 3850 . . . . . . . . . 10 dom 𝐹 ⊆ On
57 relssres 5635 . . . . . . . . . 10 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
5853, 56, 57mp2an 675 . . . . . . . . 9 (𝐹 ↾ On) = 𝐹
5958cnveqi 5492 . . . . . . . 8 (𝐹 ↾ On) = 𝐹
6059funeqi 6116 . . . . . . 7 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
6151, 60sylib 209 . . . . . 6 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun 𝐹)
6248, 61syl6 35 . . . . 5 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → Fun 𝐹))
63 onprc 7208 . . . . . 6 ¬ On ∈ V
64 funrnex 7357 . . . . . . . 8 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
6564com12 32 . . . . . . 7 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
66 df-rn 5316 . . . . . . . 8 ran 𝐹 = dom 𝐹
6766eleq1i 2872 . . . . . . 7 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
68 dfdm4 5511 . . . . . . . . 9 dom 𝐹 = ran 𝐹
6955, 68eqtr3i 2826 . . . . . . . 8 On = ran 𝐹
7069eleq1i 2872 . . . . . . 7 (On ∈ V ↔ ran 𝐹 ∈ V)
7165, 67, 703imtr4g 287 . . . . . 6 (Fun 𝐹 → (ran 𝐹 ∈ V → On ∈ V))
7263, 71mtoi 190 . . . . 5 (Fun 𝐹 → ¬ ran 𝐹 ∈ V)
7362, 72syl6 35 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ¬ ran 𝐹 ∈ V))
7438, 73jcad 504 . . 3 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
756, 74syl5bir 234 . 2 ((𝑅 Po 𝐴𝑤 We 𝐴) → (¬ ∃𝑥 ∈ On 𝐷 = ∅ → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
761, 75mt3i 143 1 ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wcel 2155  wne 2974  wral 3092  wrex 3093  {crab 3096  Vcvv 3387  wss 3763  c0 4110   class class class wbr 4837  cmpt 4916   Po wpo 5224   Or wor 5225   We wwe 5263  ccnv 5304  dom cdm 5305  ran crn 5306  cres 5307  cima 5308  Rel wrel 5310  Oncon0 5930  Fun wfun 6089   Fn wfn 6090  cfv 6095  crio 6828  recscrecs 7697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-wrecs 7636  df-recs 7698
This theorem is referenced by:  zorn2lem7  9603
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