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Theorem zornn0g 9210
 Description: Variant of Zorn's lemma zorng 9209 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 1055 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ≠ ∅)
2 simp1 1054 . . . 4 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → 𝐴 ∈ dom card)
3 snfi 7923 . . . . 5 {∅} ∈ Fin
4 finnum 8657 . . . . 5 ({∅} ∈ Fin → {∅} ∈ dom card)
53, 4ax-mp 5 . . . 4 {∅} ∈ dom card
6 unnum 8905 . . . 4 ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
72, 5, 6sylancl 693 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8 uncom 3719 . . . . . . . . 9 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
98sseq2i 3593 . . . . . . . 8 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10 ssundif 4004 . . . . . . . 8 (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
119, 10bitri 263 . . . . . . 7 (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12 difss 3699 . . . . . . . . 9 (𝑤 ∖ {∅}) ⊆ 𝑤
13 soss 4977 . . . . . . . . 9 ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅})))
1412, 13ax-mp 5 . . . . . . . 8 ( [] Or 𝑤 → [] Or (𝑤 ∖ {∅}))
15 ssdif0 3896 . . . . . . . . . . 11 (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16 uni0b 4399 . . . . . . . . . . . . 13 ( 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
1716biimpri 217 . . . . . . . . . . . 12 (𝑤 ⊆ {∅} → 𝑤 = ∅)
1817eleq1d 2672 . . . . . . . . . . 11 (𝑤 ⊆ {∅} → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
1915, 18sylbir 224 . . . . . . . . . 10 ((𝑤 ∖ {∅}) = ∅ → ( 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
2019imbi2d 329 . . . . . . . . 9 ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21 vex 3176 . . . . . . . . . . . . . . 15 𝑤 ∈ V
22 difexg 4735 . . . . . . . . . . . . . . 15 (𝑤 ∈ V → (𝑤 ∖ {∅}) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∖ {∅}) ∈ V
24 sseq1 3589 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
25 neeq1 2844 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
26 soeq2 4979 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → ( [] Or 𝑧 ↔ [] Or (𝑤 ∖ {∅})))
2724, 25, 263anbi123d 1391 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅}))))
28 unieq 4380 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤 ∖ {∅}) → 𝑧 = (𝑤 ∖ {∅}))
2928eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑧 = (𝑤 ∖ {∅}) → ( 𝑧𝐴 (𝑤 ∖ {∅}) ∈ 𝐴))
3027, 29imbi12d 333 . . . . . . . . . . . . . 14 (𝑧 = (𝑤 ∖ {∅}) → (((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [] Or (𝑤 ∖ {∅})) → (𝑤 ∖ {∅}) ∈ 𝐴)))
3123, 30spcv 3272 . . . . . . . . . . . . 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 1257 . . . . . . . . . . 11 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
3433an32s 842 . . . . . . . . . 10 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → (𝑤 ∖ {∅}) ∈ 𝐴))
35 unidif0 4764 . . . . . . . . . . . 12 (𝑤 ∖ {∅}) = 𝑤
3635eleq1i 2679 . . . . . . . . . . 11 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤𝐴)
37 elun1 3742 . . . . . . . . . . 11 ( 𝑤𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3836, 37sylbi 206 . . . . . . . . . 10 ( (𝑤 ∖ {∅}) ∈ 𝐴 𝑤 ∈ (𝐴 ∪ {∅}))
3934, 38syl6 34 . . . . . . . . 9 ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
40 0ex 4718 . . . . . . . . . . . 12 ∅ ∈ V
4140snid 4155 . . . . . . . . . . 11 ∅ ∈ {∅}
42 elun2 3743 . . . . . . . . . . 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 2867 . . . . . . . 8 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4614, 45sylan2 490 . . . . . . 7 (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [] Or 𝑤) → (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → 𝑤 ∈ (𝐴 ∪ {∅})))
4711, 46sylanb 488 . . . . . 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 1842 . . . 4 (∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
50493ad2ant3 1077 . . 3 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅})))
51 zorng 9209 . . 3 (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [] Or 𝑤) → 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
527, 50, 51syl2anc 691 . 2 ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦)
53 ssun1 3738 . . . . 5 𝐴 ⊆ (𝐴 ∪ {∅})
54 ssralv 3629 . . . . 5 (𝐴 ⊆ (𝐴 ∪ {∅}) → (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦))
5553, 54ax-mp 5 . . . 4 (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∀𝑦𝐴 ¬ 𝑥𝑦)
5655reximi 2994 . . 3 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦)
57 rexun 3755 . . . 4 (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
58 simpr 476 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
59 simpr 476 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)
60 psseq1 3656 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ⊊ 𝑦))
61 0pss 3965 . . . . . . . . . . . . 13 (∅ ⊊ 𝑦𝑦 ≠ ∅)
6260, 61syl6bb 275 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥𝑦𝑦 ≠ ∅))
6362notbid 307 . . . . . . . . . . 11 (𝑥 = ∅ → (¬ 𝑥𝑦 ↔ ¬ 𝑦 ≠ ∅))
64 nne 2786 . . . . . . . . . . 11 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
6563, 64syl6bb 275 . . . . . . . . . 10 (𝑥 = ∅ → (¬ 𝑥𝑦𝑦 = ∅))
6665ralbidv 2969 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅))
6740, 66rexsn 4170 . . . . . . . 8 (∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 𝑦 = ∅)
68 eqsn 4301 . . . . . . . . 9 (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦𝐴 𝑦 = ∅))
6968biimpar 501 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑦 = ∅) → 𝐴 = {∅})
7067, 69sylan2b 491 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → 𝐴 = {∅})
7170rexeqdv 3122 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦))
7259, 71mpbird 246 . . . . 5 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7358, 72jaodan 822 . . . 4 ((𝐴 ≠ ∅ ∧ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦𝐴 ¬ 𝑥𝑦)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7457, 73sylan2b 491 . . 3 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦𝐴 ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7556, 74sylan2 490 . 2 ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
761, 52, 75syl2anc 691 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  {csn 4125  ∪ cuni 4372   Or wor 4958  dom cdm 5038   [⊊] crpss 6834  Fincfn 7841  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rpss 6835  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-card 8648  df-cda 8873 This theorem is referenced by:  zornn0  9213  pgpfac1lem5  18301  lbsextlem4  18982  filssufilg  21525
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