Theorem wereucl 2952

Description: The unique minimal element of a subset of a well-ordered set.

Hypothesis

Ref	Expression
wereu.1	⊢ B ∈ V

Assertion

Ref	Expression
wereucl	⊢ ((R We A ⋀ B ⊆ A ⋀ B ≠ ∅) → ∪{x ∈ B∣∀y ∈ B ¬ yRx} ∈ B)

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

Proof of Theorem wereucl

Step	Hyp	Ref	Expression
1		wereu.1	. . 3 ⊢ B ∈ V
2	1	wereu 2951	. 2 ⊢ ((R We A ⋀ B ⊆ A ⋀ B ≠ ∅) → ∃!x ∈ B ∀y ∈ B ¬ yRx)
3		reucl 2891	. 2 ⊢ (∃!x ∈ B ∀y ∈ B ¬ yRx → ∪{x ∈ B∣∀y ∈ B ¬ yRx} ∈ B)
4	2, 3	syl 10	1 ⊢ ((R We A ⋀ B ⊆ A ⋀ B ≠ ∅) → ∪{x ∈ B∣∀y ∈ B ¬ yRx} ∈ B)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ w3a 777   ∈ wcel 960   ≠ wne 1588  ∀wral 1648  ∃!wreu 1650  {crab 1651  Vcvv 1814   ⊆ wss 2050  ∅c0 2283  ∪cuni 2507   class class class wbr 2624   We wwe 2922

This theorem is referenced by:  htalem 4737  zorn2lem1 4798  acdc3lem 7487  acdc2lem1 7489  acdc5lem1 7492  acdclem 7495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-po 2846  df-so 2856  df-fr 2923  df-we 2940

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