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Theorem zorn2lem4 9523
Description: Lemma for zorn2 9530. (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 5240 . . . . . . . . 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 7256 . . . . . . . 8 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
118, 9, 10sylancl 574 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12 zorn2lem.3 . . . . . . . . . . 11 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
1312tfr1 7646 . . . . . . . . . 10 𝐹 Fn On
14 fvelrnb 6385 . . . . . . . . . 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 3836 . . . . . . . . . . . . . . . . . 18 {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧} ⊆ 𝐴
2220, 21eqsstri 3784 . . . . . . . . . . . . . . . . 17 𝐷𝐴
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
2412, 23, 20zorn2lem1 9520 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
2522, 24sseldi 3750 . . . . . . . . . . . . . . . 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 3758 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹𝐴)
3611, 35ssexd 4939 . . . . . 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 9522 . . . . . . . . . . . . . 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 3773 . . . . . . . 8 On ⊆ On
5013tz7.48lem 7689 . . . . . . . 8 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
5149, 50mpan 670 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun (𝐹 ↾ On))
52 fnrel 6129 . . . . . . . . . . 11 (𝐹 Fn On → Rel 𝐹)
5313, 52ax-mp 5 . . . . . . . . . 10 Rel 𝐹
54 fndm 6130 . . . . . . . . . . . 12 (𝐹 Fn On → dom 𝐹 = On)
5513, 54ax-mp 5 . . . . . . . . . . 11 dom 𝐹 = On
5655eqimssi 3808 . . . . . . . . . 10 dom 𝐹 ⊆ On
57 relssres 5578 . . . . . . . . . 10 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
5853, 56, 57mp2an 672 . . . . . . . . 9 (𝐹 ↾ On) = 𝐹
5958cnveqi 5435 . . . . . . . 8 (𝐹 ↾ On) = 𝐹
6059funeqi 6052 . . . . . . 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 7131 . . . . . 6 ¬ On ∈ V
64 funrnex 7280 . . . . . . . 8 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
6564com12 32 . . . . . . 7 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
66 df-rn 5260 . . . . . . . 8 ran 𝐹 = dom 𝐹
6766eleq1i 2841 . . . . . . 7 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
68 dfdm4 5454 . . . . . . . . 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 502 . . 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 3723  c0 4063   class class class wbr 4786  cmpt 4863   Po wpo 5168   Or wor 5169   We wwe 5207  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Rel wrel 5254  Oncon0 5866  Fun wfun 6025   Fn wfn 6026  cfv 6031  crio 6753  recscrecs 7620
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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-wrecs 7559  df-recs 7621
This theorem is referenced by:  zorn2lem7  9526
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