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Theorem wefrc 5517
 Description: A nonempty subclass of a class well-ordered by membership 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 5510 . . 3 (𝐵𝐴 → ( E We 𝐴 → E We 𝐵))
2 n0 4263 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
3 ineq2 4136 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
43eqeq1d 2803 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐵𝑥) = ∅ ↔ (𝐵𝑦) = ∅))
54rspcev 3574 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝐵𝑦) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
65ex 416 . . . . . . . 8 (𝑦𝐵 → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
76adantl 485 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
8 inss1 4158 . . . . . . . . . . 11 (𝐵𝑦) ⊆ 𝐵
9 wefr 5513 . . . . . . . . . . . . 13 ( E We 𝐵 → E Fr 𝐵)
10 vex 3447 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1110inex2 5189 . . . . . . . . . . . . . 14 (𝐵𝑦) ∈ V
1211epfrc 5509 . . . . . . . . . . . . 13 (( E Fr 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
139, 12syl3an1 1160 . . . . . . . . . . . 12 (( E We 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
14133exp 1116 . . . . . . . . . . 11 ( E We 𝐵 → ((𝐵𝑦) ⊆ 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)))
158, 14mpi 20 . . . . . . . . . 10 ( E We 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅))
16 rexin 4169 . . . . . . . . . 10 (∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅ ↔ ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅))
1715, 16syl6ib 254 . . . . . . . . 9 ( E We 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
1817adantr 484 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
19 elin 3900 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐵𝑥) ↔ (𝑧𝐵𝑧𝑥))
20 df-3an 1086 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵))
21 3anrot 1097 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
2220, 21bitr3i 280 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
23 wetrep 5516 . . . . . . . . . . . . . . . . . . . . . . 23 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → ((𝑧𝑥𝑥𝑦) → 𝑧𝑦))
2423expd 419 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
2522, 24sylan2b 596 . . . . . . . . . . . . . . . . . . . . 21 (( E We 𝐵 ∧ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
2625exp44 441 . . . . . . . . . . . . . . . . . . . 20 ( E We 𝐵 → (𝑦𝐵 → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦))))))
2726imp 410 . . . . . . . . . . . . . . . . . . 19 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))))
2827com34 91 . . . . . . . . . . . . . . . . . 18 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑧𝑥 → (𝑥𝐵 → (𝑥𝑦𝑧𝑦)))))
2928impd 414 . . . . . . . . . . . . . . . . 17 (( E We 𝐵𝑦𝐵) → ((𝑧𝐵𝑧𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3019, 29syl5bi 245 . . . . . . . . . . . . . . . 16 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3130imp4a 426 . . . . . . . . . . . . . . 15 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → ((𝑥𝐵𝑥𝑦) → 𝑧𝑦)))
3231com23 86 . . . . . . . . . . . . . 14 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝑧 ∈ (𝐵𝑥) → 𝑧𝑦)))
3332ralrimdv 3156 . . . . . . . . . . . . 13 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦))
34 dfss3 3906 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 ↔ ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦)
3533, 34syl6ibr 255 . . . . . . . . . . . 12 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝐵𝑥) ⊆ 𝑦))
36 dfss 3902 . . . . . . . . . . . . . . 15 ((𝐵𝑥) ⊆ 𝑦 ↔ (𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦))
37 in32 4151 . . . . . . . . . . . . . . . 16 ((𝐵𝑥) ∩ 𝑦) = ((𝐵𝑦) ∩ 𝑥)
3837eqeq2i 2814 . . . . . . . . . . . . . . 15 ((𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦) ↔ (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
3936, 38sylbb 222 . . . . . . . . . . . . . 14 ((𝐵𝑥) ⊆ 𝑦 → (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
4039eqeq1d 2803 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 → ((𝐵𝑥) = ∅ ↔ ((𝐵𝑦) ∩ 𝑥) = ∅))
4140biimprd 251 . . . . . . . . . . . 12 ((𝐵𝑥) ⊆ 𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))
4235, 41syl6 35 . . . . . . . . . . 11 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅)))
4342expd 419 . . . . . . . . . 10 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → (𝑥𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))))
4443imp4a 426 . . . . . . . . 9 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → ((𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → (𝐵𝑥) = ∅)))
4544reximdvai 3234 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → (∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅))
4618, 45syld 47 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
477, 46pm2.61dne 3076 . . . . . 6 (( E We 𝐵𝑦𝐵) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
4847ex 416 . . . . 5 ( E We 𝐵 → (𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
4948exlimdv 1934 . . . 4 ( E We 𝐵 → (∃𝑦 𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
502, 49syl5bi 245 . . 3 ( E We 𝐵 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
511, 50syl6com 37 . 2 ( E We 𝐴 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅)))
52513imp 1108 1 (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110   ∩ cin 3883   ⊆ wss 3884  ∅c0 4246   E cep 5432   Fr wfr 5479   We wwe 5481 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484 This theorem is referenced by:  tz7.5  6184  onnseq  7968  finminlem  33780
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