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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 6399 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | wefrc 5683 | . 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 1537 ≠ wne 2938 ∃wrex 3068 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 E cep 5588 We wwe 5640 Ord word 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: tz7.7 6412 onint 7810 tfi 7874 peano5 7916 fin23lem26 10363 |
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