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Theorem zorn2lem4 9524
Description: Lemma for zorn2 9531. (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 pm3.24 389 . 2 ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2 df-ne 2944 . . . . 5 (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
32ralbii 3129 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4 df-ral 3066 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5 ralnex 3141 . . . 4 (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
63, 4, 53bitr3i 290 . . 3 (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7 weso 5241 . . . . . . . . 9 (𝑤 We 𝐴𝑤 Or 𝐴)
87adantr 466 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9 vex 3354 . . . . . . . 8 𝑤 ∈ V
10 soex 7257 . . . . . . . 8 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
118, 9, 10sylancl 568 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12 zorn2lem.3 . . . . . . . . . . 11 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
1312tfr1 7647 . . . . . . . . . 10 𝐹 Fn On
14 fvelrnb 6386 . . . . . . . . . 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 1995 . . . . . . . . . . 11 𝑥 𝑤 We 𝐴
17 nfa1 2184 . . . . . . . . . . 11 𝑥𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
1816, 17nfan 1980 . . . . . . . . . 10 𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19 nfv 1995 . . . . . . . . . 10 𝑥 𝑦𝐴
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
21 ssrab2 3837 . . . . . . . . . . . . . . . . . 18 {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧} ⊆ 𝐴
2220, 21eqsstri 3785 . . . . . . . . . . . . . . . . 17 𝐷𝐴
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
2412, 23, 20zorn2lem1 9521 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
2522, 24sseldi 3751 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐴)
26 eleq1 2838 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2725, 26syl5ibcom 235 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2827exp32 407 . . . . . . . . . . . . . 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 2211 . . . . . . . . . . 11 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3231imp 393 . . . . . . . . . 10 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
3318, 19, 32rexlimd 3174 . . . . . . . . 9 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹𝑥) = 𝑦𝑦𝐴))
3415, 33syl5bi 232 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
3534ssrdv 3759 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹𝐴)
3611, 35ssexd 4940 . . . . . 6 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
3736ex 397 . . . . 5 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3837adantl 467 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3912, 23, 20zorn2lem3 9523 . . . . . . . . . . . . . 14 ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
4039exp45 425 . . . . . . . . . . . . 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 393 . . . . . . . . . . 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 409 . . . . . . . . 9 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4544alrimdv 2009 . . . . . . . 8 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4645alimdv 1997 . . . . . . 7 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
47 r2al 3088 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
4846, 47syl6ibr 242 . . . . . 6 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)))
49 ssid 3774 . . . . . . . 8 On ⊆ On
5013tz7.48lem 7690 . . . . . . . 8 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
5149, 50mpan 664 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun (𝐹 ↾ On))
52 fnrel 6130 . . . . . . . . . . 11 (𝐹 Fn On → Rel 𝐹)
5313, 52ax-mp 5 . . . . . . . . . 10 Rel 𝐹
54 fndm 6131 . . . . . . . . . . . 12 (𝐹 Fn On → dom 𝐹 = On)
5513, 54ax-mp 5 . . . . . . . . . . 11 dom 𝐹 = On
5655eqimssi 3809 . . . . . . . . . 10 dom 𝐹 ⊆ On
57 relssres 5579 . . . . . . . . . 10 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
5853, 56, 57mp2an 666 . . . . . . . . 9 (𝐹 ↾ On) = 𝐹
5958cnveqi 5436 . . . . . . . 8 (𝐹 ↾ On) = 𝐹
6059funeqi 6053 . . . . . . 7 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
6151, 60sylib 208 . . . . . 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 7132 . . . . . 6 ¬ On ∈ V
64 funrnex 7281 . . . . . . . 8 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
6564com12 32 . . . . . . 7 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
66 df-rn 5261 . . . . . . . 8 ran 𝐹 = dom 𝐹
6766eleq1i 2841 . . . . . . 7 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
68 dfdm4 5455 . . . . . . . . 9 dom 𝐹 = ran 𝐹
6955, 68eqtr3i 2795 . . . . . . . 8 On = ran 𝐹
7069eleq1i 2841 . . . . . . 7 (On ∈ V ↔ ran 𝐹 ∈ V)
7165, 67, 703imtr4g 285 . . . . . 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 498 . . 3 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
756, 74syl5bir 233 . 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 196  wa 382  wal 1629   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  wss 3724  c0 4064   class class class wbr 4787  cmpt 4864   Po wpo 5169   Or wor 5170   We wwe 5208  ccnv 5249  dom cdm 5250  ran crn 5251  cres 5252  cima 5253  Rel wrel 5255  Oncon0 5867  Fun wfun 6026   Fn wfn 6027  cfv 6032  crio 6754  recscrecs 7621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-wrecs 7560  df-recs 7622
This theorem is referenced by:  zorn2lem7  9527
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