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Theorem zornn0g 10529
Description: Variant of Zorn's lemma zorng 10528 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
zornn0g ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem zornn0g
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simp2 1135 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ≠ ∅)
2 simp1 1134 . . . 4 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ∈ dom card)
3 snfi 9069 . . . . 5 {∅} ∈ Fin
4 finnum 9972 . . . . 5 ({∅} ∈ Fin → {∅} ∈ dom card)
53, 4ax-mp 5 . . . 4 {∅} ∈ dom card
6 unnum 10220 . . . 4 ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
72, 5, 6sylancl 585 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8 uncom 4152 . . . . . . . . 9 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
98sseq2i 4009 . . . . . . . 8 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10 ssundif 4488 . . . . . . . 8 (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
119, 10bitri 275 . . . . . . 7 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12 difss 4130 . . . . . . . . 9 (𝑤 ∖ {∅}) ⊆ 𝑤
13 soss 5610 . . . . . . . . 9 ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅})))
1412, 13ax-mp 5 . . . . . . . 8 ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅}))
15 ssdif0 4364 . . . . . . . . . . 11 (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16 uni0b 4936 . . . . . . . . . . . . 13 ( 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
1716biimpri 227 . . . . . . . . . . . 12 (𝑤 ⊆ {∅} → 𝑤 = ∅)
1817eleq1d 2814 . . . . . . . . . . 11 (𝑤 ⊆ {∅} → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
1915, 18sylbir 234 . . . . . . . . . 10 ((𝑤 ∖ {∅}) = ∅ → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
2019imbi2d 340 . . . . . . . . 9 ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21 vex 3475 . . . . . . . . . . . . . . 15 𝑤 ∈ V
2221difexi 5330 . . . . . . . . . . . . . 14 (𝑤 ∖ {∅}) ∈ V
23 sseq1 4005 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24 neeq1 3000 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25 soeq2 5612 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → ( [] Or 𝑧 ↔ [] Or (𝑤 ∖ {∅})))
2623, 24, 253anbi123d 1433 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅}))))
27 unieq 4919 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → 𝑧 = (𝑤 ∖ {∅}))
2827eleq1d 2814 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ( 𝑧𝐴 (𝑤 ∖ {∅}) ∈ 𝐴))
2926, 28imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = (𝑤 ∖ {∅}) → (((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (𝑤 ∖ {∅}) ∈ 𝐴)))
3022, 29spcv 3592 . . . . . . . . . . . . 13 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (𝑤 ∖ {∅}) ∈ 𝐴))
3130com12 32 . . . . . . . . . . . 12 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
32313expa 1116 . . . . . . . . . . 11 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
3332an32s 651 . . . . . . . . . 10 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
34 unidif0 5360 . . . . . . . . . . . 12 (𝑤 ∖ {∅}) = 𝑤
3534eleq1i 2820 . . . . . . . . . . 11 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤𝐴)
36 elun1 4176 . . . . . . . . . . 11 ( 𝑤𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3735, 36sylbi 216 . . . . . . . . . 10 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3833, 37syl6 35 . . . . . . . . 9 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
39 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
4039snid 4665 . . . . . . . . . . 11 ∅ ∈ {∅}
41 elun2 4177 . . . . . . . . . . 11 (∅ ∈ {∅} → ∅ ∈ (𝐴 ∪ {∅}))
4240, 41ax-mp 5 . . . . . . . . . 10 ∅ ∈ (𝐴 ∪ {∅})
43422a1i 12 . . . . . . . . 9 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅})))
4420, 38, 43pm2.61ne 3024 . . . . . . . 8 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4514, 44sylan2 592 . . . . . . 7 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4611, 45sylanb 580 . . . . . 6 ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4746com12 32 . . . . 5 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
4847alrimiv 1923 . . . 4 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
49483ad2ant3 1133 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
50 zorng 10528 . . 3 (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
517, 49, 50syl2anc 583 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
52 ssun1 4172 . . . . 5 𝐴 ⊆ (𝐴 ∪ {∅})
53 ssralv 4048 . . . . 5 (𝐴 ⊆ (𝐴 ∪ {∅}) → (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦))
5452, 53ax-mp 5 . . . 4 (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦)
5554reximi 3081 . . 3 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦)
56 rexun 4190 . . . 4 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
57 simpr 484 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
58 simpr 484 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)
59 psseq1 4085 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊊ 𝑦))
60 0pss 4445 . . . . . . . . . . . . 13 (∅ ⊊ 𝑦𝑦 ≠ ∅)
6159, 60bitrdi 287 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥𝑦𝑦 ≠ ∅))
6261notbid 318 . . . . . . . . . . 11 (𝑥 = ∅ → (¬ 𝑥𝑦 ↔ ¬ 𝑦 ≠ ∅))
63 nne 2941 . . . . . . . . . . 11 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
6462, 63bitrdi 287 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝑦𝑦 = ∅))
6564ralbidv 3174 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅))
6639, 65rexsn 4687 . . . . . . . 8 (∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅)
67 eqsn 4833 . . . . . . . . 9 (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦𝐴 𝑦 = ∅))
6867biimpar 477 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑦 = ∅) → 𝐴 = {∅})
6966, 68sylan2b 593 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → 𝐴 = {∅})
7069rexeqdv 3323 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
7158, 70mpbird 257 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7257, 71jaodan 956 . . . 4 ((𝐴 ≠ ∅ ∧ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7356, 72sylan2b 593 . . 3 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7455, 73sylan2 592 . 2 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
751, 51, 74syl2anc 583 1 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085  wal 1532   = wceq 1534  wcel 2099  wne 2937  wral 3058  wrex 3067  cdif 3944  cun 3945  wss 3947  wpss 3948  c0 4323  {csn 4629   cuni 4908   Or wor 5589  dom cdm 5678   [] crpss 7727  Fincfn 8964  cardccrd 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7728  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-en 8965  df-dom 8966  df-fin 8968  df-dju 9925  df-card 9963
This theorem is referenced by:  zornn0  10532  pgpfac1lem5  20036  lbsextlem4  21049  filssufilg  23828
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