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Theorem zornn0g 9612
Description: Variant of Zorn's lemma zorng 9611 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 1160 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ≠ ∅)
2 simp1 1159 . . . 4 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ∈ dom card)
3 snfi 8277 . . . . 5 {∅} ∈ Fin
4 finnum 9057 . . . . 5 ({∅} ∈ Fin → {∅} ∈ dom card)
53, 4ax-mp 5 . . . 4 {∅} ∈ dom card
6 unnum 9307 . . . 4 ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
72, 5, 6sylancl 576 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8 uncom 3956 . . . . . . . . 9 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
98sseq2i 3827 . . . . . . . 8 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10 ssundif 4248 . . . . . . . 8 (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
119, 10bitri 266 . . . . . . 7 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12 difss 3936 . . . . . . . . 9 (𝑤 ∖ {∅}) ⊆ 𝑤
13 soss 5250 . . . . . . . . 9 ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅})))
1412, 13ax-mp 5 . . . . . . . 8 ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅}))
15 ssdif0 4143 . . . . . . . . . . 11 (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16 uni0b 4657 . . . . . . . . . . . . 13 ( 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
1716biimpri 219 . . . . . . . . . . . 12 (𝑤 ⊆ {∅} → 𝑤 = ∅)
1817eleq1d 2870 . . . . . . . . . . 11 (𝑤 ⊆ {∅} → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
1915, 18sylbir 226 . . . . . . . . . 10 ((𝑤 ∖ {∅}) = ∅ → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
2019imbi2d 331 . . . . . . . . 9 ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21 vex 3394 . . . . . . . . . . . . . . 15 𝑤 ∈ V
22 difexg 5003 . . . . . . . . . . . . . . 15 (𝑤 ∈ V → (𝑤 ∖ {∅}) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∖ {∅}) ∈ V
24 sseq1 3823 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
25 neeq1 3040 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
26 soeq2 5252 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → ( [] Or 𝑧 ↔ [] Or (𝑤 ∖ {∅})))
2724, 25, 263anbi123d 1553 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅}))))
28 unieq 4638 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → 𝑧 = (𝑤 ∖ {∅}))
2928eleq1d 2870 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ( 𝑧𝐴 (𝑤 ∖ {∅}) ∈ 𝐴))
3027, 29imbi12d 335 . . . . . . . . . . . . . 14 (𝑧 = (𝑤 ∖ {∅}) → (((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (𝑤 ∖ {∅}) ∈ 𝐴)))
3123, 30spcv 3492 . . . . . . . . . . . . 13 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (𝑤 ∖ {∅}) ∈ 𝐴))
3231com12 32 . . . . . . . . . . . 12 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
33323expa 1140 . . . . . . . . . . 11 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
3433an32s 634 . . . . . . . . . 10 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
35 unidif0 5030 . . . . . . . . . . . 12 (𝑤 ∖ {∅}) = 𝑤
3635eleq1i 2876 . . . . . . . . . . 11 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤𝐴)
37 elun1 3979 . . . . . . . . . . 11 ( 𝑤𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3836, 37sylbi 208 . . . . . . . . . 10 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3934, 38syl6 35 . . . . . . . . 9 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
40 0ex 4984 . . . . . . . . . . . 12 ∅ ∈ V
4140snid 4402 . . . . . . . . . . 11 ∅ ∈ {∅}
42 elun2 3980 . . . . . . . . . . 11 (∅ ∈ {∅} → ∅ ∈ (𝐴 ∪ {∅}))
4341, 42ax-mp 5 . . . . . . . . . 10 ∅ ∈ (𝐴 ∪ {∅})
44432a1i 12 . . . . . . . . 9 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅})))
4520, 39, 44pm2.61ne 3063 . . . . . . . 8 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4614, 45sylan2 582 . . . . . . 7 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4711, 46sylanb 572 . . . . . 6 ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4847com12 32 . . . . 5 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
4948alrimiv 2018 . . . 4 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
50493ad2ant3 1158 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
51 zorng 9611 . . 3 (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
527, 50, 51syl2anc 575 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
53 ssun1 3975 . . . . 5 𝐴 ⊆ (𝐴 ∪ {∅})
54 ssralv 3863 . . . . 5 (𝐴 ⊆ (𝐴 ∪ {∅}) → (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦))
5553, 54ax-mp 5 . . . 4 (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦)
5655reximi 3198 . . 3 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦)
57 rexun 3992 . . . 4 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
58 simpr 473 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
59 simpr 473 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)
60 psseq1 3892 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊊ 𝑦))
61 0pss 4211 . . . . . . . . . . . . 13 (∅ ⊊ 𝑦𝑦 ≠ ∅)
6260, 61syl6bb 278 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥𝑦𝑦 ≠ ∅))
6362notbid 309 . . . . . . . . . . 11 (𝑥 = ∅ → (¬ 𝑥𝑦 ↔ ¬ 𝑦 ≠ ∅))
64 nne 2982 . . . . . . . . . . 11 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
6563, 64syl6bb 278 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝑦𝑦 = ∅))
6665ralbidv 3174 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅))
6740, 66rexsn 4416 . . . . . . . 8 (∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅)
68 eqsn 4550 . . . . . . . . 9 (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦𝐴 𝑦 = ∅))
6968biimpar 465 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑦 = ∅) → 𝐴 = {∅})
7067, 69sylan2b 583 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → 𝐴 = {∅})
7170rexeqdv 3334 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
7259, 71mpbird 248 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7358, 72jaodan 971 . . . 4 ((𝐴 ≠ ∅ ∧ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7457, 73sylan2b 583 . . 3 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7556, 74sylan2 582 . 2 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
761, 52, 75syl2anc 575 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 197  wa 384  wo 865  w3a 1100  wal 1635   = wceq 1637  wcel 2156  wne 2978  wral 3096  wrex 3097  Vcvv 3391  cdif 3766  cun 3767  wss 3769  wpss 3770  c0 4116  {csn 4370   cuni 4630   Or wor 5231  dom cdm 5311   [] crpss 7166  Fincfn 8192  cardccrd 9044
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 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-rpss 7167  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-en 8193  df-dom 8194  df-fin 8196  df-card 9048  df-cda 9275
This theorem is referenced by:  zornn0  9615  pgpfac1lem5  18680  lbsextlem4  19370  filssufilg  21928
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