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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 6195 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | wefrc 5529 | . 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) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ≠ wne 2935 ∃wrex 3055 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 E cep 5443 We wwe 5492 Ord word 6181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-ord 6185 |
This theorem is referenced by: tz7.7 6208 onint 7541 tfi 7599 peano5 7636 peano5OLD 7637 fin23lem26 9837 |
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