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Theorem wefrc 5437
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 5430 . . 3 (𝐵𝐴 → ( E We 𝐴 → E We 𝐵))
2 n0 4230 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
3 ineq2 4103 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
43eqeq1d 2797 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐵𝑥) = ∅ ↔ (𝐵𝑦) = ∅))
54rspcev 3559 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝐵𝑦) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
65ex 413 . . . . . . . 8 (𝑦𝐵 → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
76adantl 482 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) = ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
8 inss1 4125 . . . . . . . . . . 11 (𝐵𝑦) ⊆ 𝐵
9 wefr 5433 . . . . . . . . . . . . 13 ( E We 𝐵 → E Fr 𝐵)
10 vex 3440 . . . . . . . . . . . . . . 15 𝑦 ∈ V
1110inex2 5113 . . . . . . . . . . . . . 14 (𝐵𝑦) ∈ V
1211epfrc 5429 . . . . . . . . . . . . 13 (( E Fr 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
139, 12syl3an1 1156 . . . . . . . . . . . 12 (( E We 𝐵 ∧ (𝐵𝑦) ⊆ 𝐵 ∧ (𝐵𝑦) ≠ ∅) → ∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅)
14133exp 1112 . . . . . . . . . . 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 4136 . . . . . . . . . 10 (∃𝑥 ∈ (𝐵𝑦)((𝐵𝑦) ∩ 𝑥) = ∅ ↔ ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅))
1715, 16syl6ib 252 . . . . . . . . 9 ( E We 𝐵 → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
1817adantr 481 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅)))
19 elin 4090 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐵𝑥) ↔ (𝑧𝐵𝑧𝑥))
20 df-3an 1082 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵))
21 3anrot 1093 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝐵𝑧𝐵𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
2220, 21bitr3i 278 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵) ↔ (𝑧𝐵𝑥𝐵𝑦𝐵))
23 wetrep 5436 . . . . . . . . . . . . . . . . . . . . . . 23 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → ((𝑧𝑥𝑥𝑦) → 𝑧𝑦))
2423expd 416 . . . . . . . . . . . . . . . . . . . . . 22 (( E We 𝐵 ∧ (𝑧𝐵𝑥𝐵𝑦𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
2522, 24sylan2b 593 . . . . . . . . . . . . . . . . . . . . 21 (( E We 𝐵 ∧ ((𝑦𝐵𝑧𝐵) ∧ 𝑥𝐵)) → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))
2625exp44 438 . . . . . . . . . . . . . . . . . . . 20 ( E We 𝐵 → (𝑦𝐵 → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦))))))
2726imp 407 . . . . . . . . . . . . . . . . . . 19 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑥𝐵 → (𝑧𝑥 → (𝑥𝑦𝑧𝑦)))))
2827com34 91 . . . . . . . . . . . . . . . . . 18 (( E We 𝐵𝑦𝐵) → (𝑧𝐵 → (𝑧𝑥 → (𝑥𝐵 → (𝑥𝑦𝑧𝑦)))))
2928impd 411 . . . . . . . . . . . . . . . . 17 (( E We 𝐵𝑦𝐵) → ((𝑧𝐵𝑧𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3019, 29syl5bi 243 . . . . . . . . . . . . . . . 16 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → (𝑥𝐵 → (𝑥𝑦𝑧𝑦))))
3130imp4a 423 . . . . . . . . . . . . . . 15 (( E We 𝐵𝑦𝐵) → (𝑧 ∈ (𝐵𝑥) → ((𝑥𝐵𝑥𝑦) → 𝑧𝑦)))
3231com23 86 . . . . . . . . . . . . . 14 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝑧 ∈ (𝐵𝑥) → 𝑧𝑦)))
3332ralrimdv 3155 . . . . . . . . . . . . 13 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦))
34 dfss3 3878 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 ↔ ∀𝑧 ∈ (𝐵𝑥)𝑧𝑦)
3533, 34syl6ibr 253 . . . . . . . . . . . 12 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (𝐵𝑥) ⊆ 𝑦))
36 dfss 3875 . . . . . . . . . . . . . . 15 ((𝐵𝑥) ⊆ 𝑦 ↔ (𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦))
37 in32 4118 . . . . . . . . . . . . . . . 16 ((𝐵𝑥) ∩ 𝑦) = ((𝐵𝑦) ∩ 𝑥)
3837eqeq2i 2807 . . . . . . . . . . . . . . 15 ((𝐵𝑥) = ((𝐵𝑥) ∩ 𝑦) ↔ (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
3936, 38sylbb 220 . . . . . . . . . . . . . 14 ((𝐵𝑥) ⊆ 𝑦 → (𝐵𝑥) = ((𝐵𝑦) ∩ 𝑥))
4039eqeq1d 2797 . . . . . . . . . . . . 13 ((𝐵𝑥) ⊆ 𝑦 → ((𝐵𝑥) = ∅ ↔ ((𝐵𝑦) ∩ 𝑥) = ∅))
4140biimprd 249 . . . . . . . . . . . 12 ((𝐵𝑥) ⊆ 𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))
4235, 41syl6 35 . . . . . . . . . . 11 (( E We 𝐵𝑦𝐵) → ((𝑥𝐵𝑥𝑦) → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅)))
4342expd 416 . . . . . . . . . 10 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → (𝑥𝑦 → (((𝐵𝑦) ∩ 𝑥) = ∅ → (𝐵𝑥) = ∅))))
4443imp4a 423 . . . . . . . . 9 (( E We 𝐵𝑦𝐵) → (𝑥𝐵 → ((𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → (𝐵𝑥) = ∅)))
4544reximdvai 3235 . . . . . . . 8 (( E We 𝐵𝑦𝐵) → (∃𝑥𝐵 (𝑥𝑦 ∧ ((𝐵𝑦) ∩ 𝑥) = ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅))
4618, 45syld 47 . . . . . . 7 (( E We 𝐵𝑦𝐵) → ((𝐵𝑦) ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
477, 46pm2.61dne 3071 . . . . . 6 (( E We 𝐵𝑦𝐵) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
4847ex 413 . . . . 5 ( E We 𝐵 → (𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
4948exlimdv 1911 . . . 4 ( E We 𝐵 → (∃𝑦 𝑦𝐵 → ∃𝑥𝐵 (𝐵𝑥) = ∅))
502, 49syl5bi 243 . . 3 ( E We 𝐵 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅))
511, 50syl6com 37 . 2 ( E We 𝐴 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵 (𝐵𝑥) = ∅)))
52513imp 1104 1 (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  wne 2984  wral 3105  wrex 3106  cin 3858  wss 3859  c0 4211   E cep 5352   Fr wfr 5399   We wwe 5401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404
This theorem is referenced by:  tz7.5  6087  onnseq  7833  finminlem  33275
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