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Theorem wefrc 5137
Description: A nonempty (possibly proper) subclass of a class well-ordered by E has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
wefrc (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem wefrc
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wess 5130 . . 3 (𝐵𝐴 → ( E We 𝐴 → E We 𝐵))
2 n0 3964 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
3 ineq2 3841 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
43eqeq1d 2653 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐵𝑥) = ∅ ↔ (𝐵𝑦) = ∅))
54rspcev 3340 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝐵𝑦) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
65ex 449 . . . . . . . 8 (𝑦𝐵 → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
76adantl 481 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
8 inss1 3866 . . . . . . . . . . 11 (𝐵𝑦) ⊆ 𝐵
9 wefr 5133 . . . . . . . . . . . . 13 ( E We 𝐵 → E Fr 𝐵)
10 vex 3234 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1110inex2 4833 . . . . . . . . . . . . . 14 (𝐵𝑦) ∈ V
1211epfrc 5129 . . . . . . . . . . . . 13 (( E Fr 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
139, 12syl3an1 1399 . . . . . . . . . . . 12 (( E We 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
14133exp 1283 . . . . . . . . . . 11 ( E We 𝐵 → ((𝐵𝑦) ⊆ 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)))
158, 14mpi 20 . . . . . . . . . 10 ( E We 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅))
16 elin 3829 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵𝑦) ↔ (𝑥𝐵𝑥𝑦))
1716anbi1i 731 . . . . . . . . . . . 12 ((𝑥 ∈ (𝐵𝑦) ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) ↔ ((𝑥𝐵𝑥𝑦) ∧ ((𝐵𝑦) ∩ 𝑥) = ∅))
18 anass 682 . . . . . . . . . . . 12 (((𝑥𝐵𝑥𝑦) ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) ↔ (𝑥𝐵 ∧ (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
1917, 18bitri 264 . . . . . . . . . . 11 ((𝑥 ∈ (𝐵𝑦) ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) ↔ (𝑥𝐵 ∧ (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
2019rexbii2 3068 . . . . . . . . . 10 (∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅ ↔ ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅))
2115, 20syl6ib 241 . . . . . . . . 9 ( E We 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
2221adantr 480 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
23 elin 3829 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐵𝑥) ↔ (𝑧𝐵𝑧𝑥))
24 df-3an 1056 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵))
25 3anrot 1060 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
2624, 25bitr3i 266 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
27 wetrep 5136 . . . . . . . . . . . . . . . . . . . . . . 23 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → ((𝑧𝑥𝑥𝑦) → 𝑧𝑦))
2827expd 451 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
2926, 28sylan2b 491 . . . . . . . . . . . . . . . . . . . . 21 (( E We 𝐵 ∧ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
3029exp44 640 . . . . . . . . . . . . . . . . . . . 20 ( E We 𝐵 → (𝑦𝐵 → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦))))))
3130imp 444 . . . . . . . . . . . . . . . . . . 19 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))))
3231com34 91 . . . . . . . . . . . . . . . . . 18 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑧𝑥 → (𝑥𝐵 → (𝑥𝑦𝑧𝑦)))))
3332impd 446 . . . . . . . . . . . . . . . . 17 (( E We 𝐵𝑦𝐵) → ((𝑧𝐵𝑧𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3423, 33syl5bi 232 . . . . . . . . . . . . . . . 16 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3534imp4a 613 . . . . . . . . . . . . . . 15 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → ((𝑥𝐵𝑥𝑦) → 𝑧𝑦)))
3635com23 86 . . . . . . . . . . . . . 14 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝑧 ∈ (𝐵𝑥) → 𝑧𝑦)))
3736ralrimdv 2997 . . . . . . . . . . . . 13 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦))
38 dfss3 3625 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 ↔ ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦)
3937, 38syl6ibr 242 . . . . . . . . . . . 12 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝐵𝑥) ⊆ 𝑦))
40 dfss 3622 . . . . . . . . . . . . . . 15 ((𝐵𝑥) ⊆ 𝑦 ↔ (𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦))
41 in32 3858 . . . . . . . . . . . . . . . 16 ((𝐵𝑥) ∩ 𝑦) = ((𝐵𝑦) ∩ 𝑥)
4241eqeq2i 2663 . . . . . . . . . . . . . . 15 ((𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦) ↔ (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
4340, 42sylbb 209 . . . . . . . . . . . . . 14 ((𝐵𝑥) ⊆ 𝑦 → (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
4443eqeq1d 2653 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 → ((𝐵𝑥) = ∅ ↔ ((𝐵𝑦) ∩ 𝑥) = ∅))
4544biimprd 238 . . . . . . . . . . . 12 ((𝐵𝑥) ⊆ 𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))
4639, 45syl6 35 . . . . . . . . . . 11 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅)))
4746expd 451 . . . . . . . . . 10 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → (𝑥𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))))
4847imp4a 613 . . . . . . . . 9 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → ((𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → (𝐵𝑥) = ∅)))
4948reximdvai 3044 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → (∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅))
5022, 49syld 47 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
517, 50pm2.61dne 2909 . . . . . 6 (( E We 𝐵𝑦𝐵) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
5251ex 449 . . . . 5 ( E We 𝐵 → (𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
5352exlimdv 1901 . . . 4 ( E We 𝐵 → (∃𝑦 𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
542, 53syl5bi 232 . . 3 ( E We 𝐵 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
551, 54syl6com 37 . 2 ( E We 𝐴 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅)))
56553imp 1275 1 (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948   E cep 5057   Fr wfr 5099   We wwe 5101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104
This theorem is referenced by:  tz7.5  5782  onnseq  7486  finminlem  32437
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