Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > entri3 | Structured version Visualization version GIF version |
Description: Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.) |
Ref | Expression |
---|---|
entri3 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | entri2 10051 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≺ 𝐴)) | |
2 | sdomdom 8576 | . . 3 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | 2 | orim2i 910 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∨ 𝐵 ≺ 𝐴) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) |
4 | 1, 3 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 ∈ wcel 2113 class class class wbr 5027 ≼ cdom 8546 ≺ csdm 8547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-ac2 9956 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-wrecs 7969 df-recs 8030 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-card 9434 df-ac 9609 |
This theorem is referenced by: alephval2 10065 |
Copyright terms: Public domain | W3C validator |