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Mirrors > Home > MPE Home > Th. List > onnmin | Structured version Visualization version GIF version |
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) |
Ref | Expression |
---|---|
onnmin | ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 4968 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵) | |
2 | 1 | adantl 483 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ∩ 𝐴 ⊆ 𝐵) |
3 | ne0i 4335 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) | |
4 | oninton 7783 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On) | |
5 | 3, 4 | sylan2 594 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ∩ 𝐴 ∈ On) |
6 | ssel2 3978 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
7 | ontri1 6399 | . . 3 ⊢ ((∩ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (∩ 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ∩ 𝐴)) | |
8 | 5, 6, 7 | syl2anc 585 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → (∩ 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ∩ 𝐴)) |
9 | 2, 8 | mpbid 231 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3949 ∅c0 4323 ∩ cint 4951 Oncon0 6365 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: onnminsb 7787 oneqmin 7788 onmindif2 7795 cardmin2 9994 ackbij1lem18 10232 cofsmo 10264 fin23lem26 10320 lrrecfr 27427 |
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