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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1071 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1071.7 | ⊢ 𝐷 = (ω ∖ {∅}) |
Ref | Expression |
---|---|
bnj1071 | ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1071.7 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
2 | 1 | bnj923 30838 | . 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
3 | nnord 7073 | . 2 ⊢ (𝑛 ∈ ω → Ord 𝑛) | |
4 | ordfr 5738 | . 2 ⊢ (Ord 𝑛 → E Fr 𝑛) | |
5 | 2, 3, 4 | 3syl 18 | 1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∅c0 3915 {csn 4177 E cep 5028 Fr wfr 5070 Ord word 5722 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: bnj1030 31055 bnj1133 31057 |
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