Theorem tz6.26i 6321

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 6320	. 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 1540   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296   Se wse 5589   We wwe 5590  Predcpred 6273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274

This theorem is referenced by: (None)