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Theorem tz7.5 6404
Description: A nonempty subclass of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)
Assertion
Ref Expression
tz7.5 ((Ord 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.5
StepHypRef Expression
1 ordwe 6396 . 2 (Ord 𝐴 → E We 𝐴)
2 wefrc 5678 . 2 (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
31, 2syl3an1 1163 1 ((Ord 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wne 2939  wrex 3069  cin 3949  wss 3950  c0 4332   E cep 5582   We wwe 5635  Ord word 6382
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-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
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-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386
This theorem is referenced by:  tz7.7  6409  onint  7811  tfi  7875  peano5  7916  fin23lem26  10366
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