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Mirrors > Home > MPE Home > Th. List > tz7.5 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
tz7.5 | ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordwe 6279 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | wefrc 5583 | . 2 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) | |
3 | 1, 2 | syl3an1 1162 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ≠ wne 2943 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 E cep 5494 We wwe 5543 Ord word 6265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-ord 6269 |
This theorem is referenced by: tz7.7 6292 onint 7640 tfi 7700 peano5 7740 peano5OLD 7741 fin23lem26 10081 |
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