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Mirrors > Home > MPE Home > Th. List > nnm0 | Structured version Visualization version GIF version |
Description: Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
Ref | Expression |
---|---|
nnm0 | ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7877 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | om0 8538 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4322 Oncon0 6371 (class class class)co 7419 ωcom 7871 ·o comu 8485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-omul 8492 |
This theorem is referenced by: nnmcl 8633 nndi 8644 nnmass 8645 nnmsucr 8646 nnmcom 8647 omabs 8672 nnm1 8673 |
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