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Mirrors > Home > MPE Home > Th. List > infsdomnn | Structured version Visualization version GIF version |
Description: An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
infsdomnn | ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8249 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5408 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ V) |
3 | nnsdomg 8509 | . . 3 ⊢ ((ω ∈ V ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) | |
4 | 2, 3 | sylan 575 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) |
5 | simpl 476 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → ω ≼ 𝐴) | |
6 | sdomdomtr 8383 | . 2 ⊢ ((𝐵 ≺ ω ∧ ω ≼ 𝐴) → 𝐵 ≺ 𝐴) | |
7 | 4, 5, 6 | syl2anc 579 | 1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 Vcvv 3398 class class class wbr 4888 ωcom 7345 ≼ cdom 8241 ≺ csdm 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 |
This theorem is referenced by: infn0 8512 |
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