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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj529 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj529.1 | ⊢ 𝐷 = (ω ∖ {∅}) |
Ref | Expression |
---|---|
bnj529 | ⊢ (𝑀 ∈ 𝐷 → ∅ ∈ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4705 | . . . 4 ⊢ (𝑀 ∈ (ω ∖ {∅}) ↔ (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) | |
2 | 1 | biimpi 218 | . . 3 ⊢ (𝑀 ∈ (ω ∖ {∅}) → (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) |
3 | bnj529.1 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
4 | 2, 3 | eleq2s 2931 | . 2 ⊢ (𝑀 ∈ 𝐷 → (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) |
5 | nnord 7574 | . . 3 ⊢ (𝑀 ∈ ω → Ord 𝑀) | |
6 | 5 | anim1i 616 | . 2 ⊢ ((𝑀 ∈ ω ∧ 𝑀 ≠ ∅) → (Ord 𝑀 ∧ 𝑀 ≠ ∅)) |
7 | ord0eln0 6231 | . . 3 ⊢ (Ord 𝑀 → (∅ ∈ 𝑀 ↔ 𝑀 ≠ ∅)) | |
8 | 7 | biimpar 480 | . 2 ⊢ ((Ord 𝑀 ∧ 𝑀 ≠ ∅) → ∅ ∈ 𝑀) |
9 | 4, 6, 8 | 3syl 18 | 1 ⊢ (𝑀 ∈ 𝐷 → ∅ ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3921 ∅c0 4279 {csn 4553 Ord word 6176 ωcom 7566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-tr 5159 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-om 7567 |
This theorem is referenced by: bnj545 32174 bnj900 32208 bnj929 32215 |
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