Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
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 4686 | . . . 4 ⊢ (𝑀 ∈ (ω ∖ {∅}) ↔ (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) | |
2 | 1 | biimpi 219 | . . 3 ⊢ (𝑀 ∈ (ω ∖ {∅}) → (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) |
3 | bnj529.1 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
4 | 2, 3 | eleq2s 2849 | . 2 ⊢ (𝑀 ∈ 𝐷 → (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) |
5 | nnord 7630 | . . 3 ⊢ (𝑀 ∈ ω → Ord 𝑀) | |
6 | 5 | anim1i 618 | . 2 ⊢ ((𝑀 ∈ ω ∧ 𝑀 ≠ ∅) → (Ord 𝑀 ∧ 𝑀 ≠ ∅)) |
7 | ord0eln0 6245 | . . 3 ⊢ (Ord 𝑀 → (∅ ∈ 𝑀 ↔ 𝑀 ≠ ∅)) | |
8 | 7 | biimpar 481 | . 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 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 ∅c0 4223 {csn 4527 Ord word 6190 ωcom 7622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-om 7623 |
This theorem is referenced by: bnj545 32542 bnj900 32576 bnj929 32583 |
Copyright terms: Public domain | W3C validator |