Theorem tz7.5 6339

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 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Distinct variable group:   𝑥,𝐵

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.5

Step	Hyp	Ref	Expression
1		ordwe 6331	. 2 ⊢ (Ord 𝐴 → E We 𝐴)
2		wefrc 5619	. 2 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)
3	1, 2	syl3an1 1164	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ≠ wne 2933  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274   E cep 5524   We wwe 5577  Ord word 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321

This theorem is referenced by:  tz7.7  6344  onint  7738  tfi  7798  peano5  7838  fin23lem26  10241