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Theorem tz7.5 5782
 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 5774 . 2 (Ord 𝐴 → E We 𝐴)
2 wefrc 5137 . 2 (( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
31, 2syl3an1 1399 1 ((Ord 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ≠ wne 2823  ∃wrex 2942   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948   E cep 5057   We wwe 5101  Ord word 5760 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764 This theorem is referenced by:  tz7.7  5787  onint  7037  tfi  7095  peano5  7131  fin23lem26  9185
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