Theorem tz6.26i 6351

Description: All nonempty subclasses of a class having a well-ordered set-like relation 𝑅 have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
tz6.26i.1	⊢ 𝑅 We 𝐴
tz6.26i.2	⊢ 𝑅 Se 𝐴

Assertion

Ref	Expression
tz6.26i	⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem tz6.26i

Step	Hyp	Ref	Expression
1		tz6.26i.1	. 2 ⊢ 𝑅 We 𝐴
2		tz6.26i.2	. 2 ⊢ 𝑅 Se 𝐴
3		tz6.26 6349	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
4	1, 2, 3	mpanl12 702	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ≠ wne 2931  ∃wrex 3059   ⊆ wss 3933  ∅c0 4315   Se wse 5617   We wwe 5618  Predcpred 6302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303

This theorem is referenced by:  wfrlem16OLD  8347