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Theorem epfrs 9206
 Description: The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9207. (Contributed by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
epfrs (( E Fr 𝐴𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem epfrs
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4245 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧𝐴)
2 snssi 4698 . . . . . . . . . . . 12 (𝑧𝐴 → {𝑧} ⊆ 𝐴)
32anim2i 619 . . . . . . . . . . 11 (({𝑧} ⊆ 𝑦𝑧𝐴) → ({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴))
4 ssin 4135 . . . . . . . . . . . 12 (({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴) ↔ {𝑧} ⊆ (𝑦𝐴))
5 vex 3413 . . . . . . . . . . . . 13 𝑧 ∈ V
65snss 4676 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦𝐴) ↔ {𝑧} ⊆ (𝑦𝐴))
74, 6bitr4i 281 . . . . . . . . . . 11 (({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴) ↔ 𝑧 ∈ (𝑦𝐴))
83, 7sylib 221 . . . . . . . . . 10 (({𝑧} ⊆ 𝑦𝑧𝐴) → 𝑧 ∈ (𝑦𝐴))
98ne0d 4234 . . . . . . . . 9 (({𝑧} ⊆ 𝑦𝑧𝐴) → (𝑦𝐴) ≠ ∅)
10 inss2 4134 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝐴
11 vex 3413 . . . . . . . . . . . . . 14 𝑦 ∈ V
1211inex1 5187 . . . . . . . . . . . . 13 (𝑦𝐴) ∈ V
1312epfrc 5510 . . . . . . . . . . . 12 (( E Fr 𝐴 ∧ (𝑦𝐴) ⊆ 𝐴 ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)((𝑦𝐴) ∩ 𝑥) = ∅)
1410, 13mp3an2 1446 . . . . . . . . . . 11 (( E Fr 𝐴 ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)((𝑦𝐴) ∩ 𝑥) = ∅)
15 elin 3874 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
1615anbi1i 626 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑦𝐴) ∧ ((𝑦𝐴) ∩ 𝑥) = ∅) ↔ ((𝑥𝑦𝑥𝐴) ∧ ((𝑦𝐴) ∩ 𝑥) = ∅))
17 anass 472 . . . . . . . . . . . . . 14 (((𝑥𝑦𝑥𝐴) ∧ ((𝑦𝐴) ∩ 𝑥) = ∅) ↔ (𝑥𝑦 ∧ (𝑥𝐴 ∧ ((𝑦𝐴) ∩ 𝑥) = ∅)))
1816, 17bitri 278 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑦𝐴) ∧ ((𝑦𝐴) ∩ 𝑥) = ∅) ↔ (𝑥𝑦 ∧ (𝑥𝐴 ∧ ((𝑦𝐴) ∩ 𝑥) = ∅)))
19 n0 4245 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑥𝐴))
20 elinel1 4100 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ (𝑥𝐴) → 𝑤𝑥)
2120ancri 553 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝑥𝐴) → (𝑤𝑥𝑤 ∈ (𝑥𝐴)))
22 trel 5145 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Tr 𝑦 → ((𝑤𝑥𝑥𝑦) → 𝑤𝑦))
23 inass 4124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦𝐴) ∩ 𝑥) = (𝑦 ∩ (𝐴𝑥))
24 incom 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴𝑥) = (𝑥𝐴)
2524ineq2i 4114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∩ (𝐴𝑥)) = (𝑦 ∩ (𝑥𝐴))
2623, 25eqtri 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦𝐴) ∩ 𝑥) = (𝑦 ∩ (𝑥𝐴))
2726eleq2i 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ ((𝑦𝐴) ∩ 𝑥) ↔ 𝑤 ∈ (𝑦 ∩ (𝑥𝐴)))
28 elin 3874 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (𝑦 ∩ (𝑥𝐴)) ↔ (𝑤𝑦𝑤 ∈ (𝑥𝐴)))
2927, 28bitr2i 279 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑤𝑦𝑤 ∈ (𝑥𝐴)) ↔ 𝑤 ∈ ((𝑦𝐴) ∩ 𝑥))
30 ne0i 4233 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ ((𝑦𝐴) ∩ 𝑥) → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)
3129, 30sylbi 220 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑤𝑦𝑤 ∈ (𝑥𝐴)) → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)
3231ex 416 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤𝑦 → (𝑤 ∈ (𝑥𝐴) → ((𝑦𝐴) ∩ 𝑥) ≠ ∅))
3322, 32syl6 35 . . . . . . . . . . . . . . . . . . . . . . . 24 (Tr 𝑦 → ((𝑤𝑥𝑥𝑦) → (𝑤 ∈ (𝑥𝐴) → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
3433expd 419 . . . . . . . . . . . . . . . . . . . . . . 23 (Tr 𝑦 → (𝑤𝑥 → (𝑥𝑦 → (𝑤 ∈ (𝑥𝐴) → ((𝑦𝐴) ∩ 𝑥) ≠ ∅))))
3534com34 91 . . . . . . . . . . . . . . . . . . . . . 22 (Tr 𝑦 → (𝑤𝑥 → (𝑤 ∈ (𝑥𝐴) → (𝑥𝑦 → ((𝑦𝐴) ∩ 𝑥) ≠ ∅))))
3635impd 414 . . . . . . . . . . . . . . . . . . . . 21 (Tr 𝑦 → ((𝑤𝑥𝑤 ∈ (𝑥𝐴)) → (𝑥𝑦 → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
3721, 36syl5 34 . . . . . . . . . . . . . . . . . . . 20 (Tr 𝑦 → (𝑤 ∈ (𝑥𝐴) → (𝑥𝑦 → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
3837exlimdv 1934 . . . . . . . . . . . . . . . . . . 19 (Tr 𝑦 → (∃𝑤 𝑤 ∈ (𝑥𝐴) → (𝑥𝑦 → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
3919, 38syl5bi 245 . . . . . . . . . . . . . . . . . 18 (Tr 𝑦 → ((𝑥𝐴) ≠ ∅ → (𝑥𝑦 → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
4039com23 86 . . . . . . . . . . . . . . . . 17 (Tr 𝑦 → (𝑥𝑦 → ((𝑥𝐴) ≠ ∅ → ((𝑦𝐴) ∩ 𝑥) ≠ ∅)))
4140imp 410 . . . . . . . . . . . . . . . 16 ((Tr 𝑦𝑥𝑦) → ((𝑥𝐴) ≠ ∅ → ((𝑦𝐴) ∩ 𝑥) ≠ ∅))
4241necon4d 2975 . . . . . . . . . . . . . . 15 ((Tr 𝑦𝑥𝑦) → (((𝑦𝐴) ∩ 𝑥) = ∅ → (𝑥𝐴) = ∅))
4342anim2d 614 . . . . . . . . . . . . . 14 ((Tr 𝑦𝑥𝑦) → ((𝑥𝐴 ∧ ((𝑦𝐴) ∩ 𝑥) = ∅) → (𝑥𝐴 ∧ (𝑥𝐴) = ∅)))
4443expimpd 457 . . . . . . . . . . . . 13 (Tr 𝑦 → ((𝑥𝑦 ∧ (𝑥𝐴 ∧ ((𝑦𝐴) ∩ 𝑥) = ∅)) → (𝑥𝐴 ∧ (𝑥𝐴) = ∅)))
4518, 44syl5bi 245 . . . . . . . . . . . 12 (Tr 𝑦 → ((𝑥 ∈ (𝑦𝐴) ∧ ((𝑦𝐴) ∩ 𝑥) = ∅) → (𝑥𝐴 ∧ (𝑥𝐴) = ∅)))
4645reximdv2 3195 . . . . . . . . . . 11 (Tr 𝑦 → (∃𝑥 ∈ (𝑦𝐴)((𝑦𝐴) ∩ 𝑥) = ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅))
4714, 46syl5 34 . . . . . . . . . 10 (Tr 𝑦 → (( E Fr 𝐴 ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅))
4847expcomd 420 . . . . . . . . 9 (Tr 𝑦 → ((𝑦𝐴) ≠ ∅ → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅)))
499, 48syl5 34 . . . . . . . 8 (Tr 𝑦 → (({𝑧} ⊆ 𝑦𝑧𝐴) → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅)))
5049expd 419 . . . . . . 7 (Tr 𝑦 → ({𝑧} ⊆ 𝑦 → (𝑧𝐴 → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅))))
5150impcom 411 . . . . . 6 (({𝑧} ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧𝐴 → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅)))
52513adant3 1129 . . . . 5 (({𝑧} ⊆ 𝑦 ∧ Tr 𝑦 ∧ ∀𝑤(({𝑧} ⊆ 𝑤 ∧ Tr 𝑤) → 𝑦𝑤)) → (𝑧𝐴 → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅)))
53 snex 5300 . . . . . 6 {𝑧} ∈ V
5453tz9.1 9204 . . . . 5 𝑦({𝑧} ⊆ 𝑦 ∧ Tr 𝑦 ∧ ∀𝑤(({𝑧} ⊆ 𝑤 ∧ Tr 𝑤) → 𝑦𝑤))
5552, 54exlimiiv 1932 . . . 4 (𝑧𝐴 → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅))
5655exlimiv 1931 . . 3 (∃𝑧 𝑧𝐴 → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅))
571, 56sylbi 220 . 2 (𝐴 ≠ ∅ → ( E Fr 𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅))
5857impcom 411 1 (( E Fr 𝐴𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {csn 4522  Tr wtr 5138   E cep 5434   Fr wfr 5480 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459  ax-inf2 9137 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056 This theorem is referenced by:  zfregs  9207
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