Theorem tz6.26i 6358

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 6356	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
4	1, 2, 3	mpanl12 700	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ≠ wne 2936  ∃wrex 3066   ⊆ wss 3947  ∅c0 4324   Se wse 5633   We wwe 5634  Predcpred 6307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308

This theorem is referenced by:  wfrlem16OLD  8349