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Theorem zornn0g 9921
 Description: Variant of Zorn's lemma zorng 9920 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 1134 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ≠ ∅)
2 simp1 1133 . . . 4 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ∈ dom card)
3 snfi 8586 . . . . 5 {∅} ∈ Fin
4 finnum 9370 . . . . 5 ({∅} ∈ Fin → {∅} ∈ dom card)
53, 4ax-mp 5 . . . 4 {∅} ∈ dom card
6 unnum 9616 . . . 4 ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
72, 5, 6sylancl 589 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8 uncom 4115 . . . . . . . . 9 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
98sseq2i 3982 . . . . . . . 8 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10 ssundif 4416 . . . . . . . 8 (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
119, 10bitri 278 . . . . . . 7 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12 difss 4094 . . . . . . . . 9 (𝑤 ∖ {∅}) ⊆ 𝑤
13 soss 5481 . . . . . . . . 9 ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅})))
1412, 13ax-mp 5 . . . . . . . 8 ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅}))
15 ssdif0 4306 . . . . . . . . . . 11 (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16 uni0b 4851 . . . . . . . . . . . . 13 ( 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
1716biimpri 231 . . . . . . . . . . . 12 (𝑤 ⊆ {∅} → 𝑤 = ∅)
1817eleq1d 2900 . . . . . . . . . . 11 (𝑤 ⊆ {∅} → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
1915, 18sylbir 238 . . . . . . . . . 10 ((𝑤 ∖ {∅}) = ∅ → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
2019imbi2d 344 . . . . . . . . 9 ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21 vex 3483 . . . . . . . . . . . . . . 15 𝑤 ∈ V
2221difexi 5219 . . . . . . . . . . . . . 14 (𝑤 ∖ {∅}) ∈ V
23 sseq1 3978 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24 neeq1 3076 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25 soeq2 5483 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → ( [] Or 𝑧 ↔ [] Or (𝑤 ∖ {∅})))
2623, 24, 253anbi123d 1433 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅}))))
27 unieq 4836 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → 𝑧 = (𝑤 ∖ {∅}))
2827eleq1d 2900 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ( 𝑧𝐴 (𝑤 ∖ {∅}) ∈ 𝐴))
2926, 28imbi12d 348 . . . . . . . . . . . . . 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 1115 . . . . . . . . . . 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 5248 . . . . . . . . . . . 12 (𝑤 ∖ {∅}) = 𝑤
3534eleq1i 2906 . . . . . . . . . . 11 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤𝐴)
36 elun1 4138 . . . . . . . . . . 11 ( 𝑤𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3735, 36sylbi 220 . . . . . . . . . 10 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3833, 37syl6 35 . . . . . . . . 9 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
39 0ex 5198 . . . . . . . . . . . 12 ∅ ∈ V
4039snid 4586 . . . . . . . . . . 11 ∅ ∈ {∅}
41 elun2 4139 . . . . . . . . . . 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 3099 . . . . . . . 8 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4514, 44sylan2 595 . . . . . . 7 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4611, 45sylanb 584 . . . . . 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 1929 . . . 4 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
49483ad2ant3 1132 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
50 zorng 9920 . . 3 (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
517, 49, 50syl2anc 587 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
52 ssun1 4134 . . . . 5 𝐴 ⊆ (𝐴 ∪ {∅})
53 ssralv 4019 . . . . 5 (𝐴 ⊆ (𝐴 ∪ {∅}) → (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦))
5452, 53ax-mp 5 . . . 4 (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦)
5554reximi 3238 . . 3 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦)
56 rexun 4152 . . . 4 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
57 simpr 488 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
58 simpr 488 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)
59 psseq1 4050 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊊ 𝑦))
60 0pss 4379 . . . . . . . . . . . . 13 (∅ ⊊ 𝑦𝑦 ≠ ∅)
6159, 60syl6bb 290 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥𝑦𝑦 ≠ ∅))
6261notbid 321 . . . . . . . . . . 11 (𝑥 = ∅ → (¬ 𝑥𝑦 ↔ ¬ 𝑦 ≠ ∅))
63 nne 3018 . . . . . . . . . . 11 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
6462, 63syl6bb 290 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝑦𝑦 = ∅))
6564ralbidv 3192 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅))
6639, 65rexsn 4605 . . . . . . . 8 (∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅)
67 eqsn 4747 . . . . . . . . 9 (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦𝐴 𝑦 = ∅))
6867biimpar 481 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑦 = ∅) → 𝐴 = {∅})
6966, 68sylan2b 596 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → 𝐴 = {∅})
7069rexeqdv 3404 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
7158, 70mpbird 260 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7257, 71jaodan 955 . . . 4 ((𝐴 ≠ ∅ ∧ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7356, 72sylan2b 596 . . 3 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7455, 73sylan2 595 . 2 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
751, 51, 74syl2anc 587 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134   ∖ cdif 3916   ∪ cun 3917   ⊆ wss 3919   ⊊ wpss 3920  ∅c0 4276  {csn 4550  ∪ cuni 4825   Or wor 5461  dom cdm 5543   [⊊] crpss 7439  Fincfn 8501  cardccrd 9357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-rpss 7440  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-fin 8505  df-dju 9323  df-card 9361 This theorem is referenced by:  zornn0  9924  pgpfac1lem5  19199  lbsextlem4  19928  filssufilg  22514
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