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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 4711 | . . . 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 2929 | . 2 ⊢ (𝑀 ∈ 𝐷 → (𝑀 ∈ ω ∧ 𝑀 ≠ ∅)) |
5 | nnord 7580 | . . 3 ⊢ (𝑀 ∈ ω → Ord 𝑀) | |
6 | 5 | anim1i 616 | . 2 ⊢ ((𝑀 ∈ ω ∧ 𝑀 ≠ ∅) → (Ord 𝑀 ∧ 𝑀 ≠ ∅)) |
7 | ord0eln0 6238 | . . 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 1531 ∈ wcel 2108 ≠ wne 3014 ∖ cdif 3931 ∅c0 4289 {csn 4559 Ord word 6183 ωcom 7572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7573 |
This theorem is referenced by: bnj545 32160 bnj900 32194 bnj929 32201 |
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