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Mirrors > Home > MPE Home > Th. List > infsdomnnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of infsdomnn 9323 as of 7-Jan-2025. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
infsdomnnOLD | ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8963 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5729 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ V) |
3 | nnsdomg 9320 | . . 3 ⊢ ((ω ∈ V ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) | |
4 | 2, 3 | sylan 578 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) |
5 | simpl 481 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → ω ≼ 𝐴) | |
6 | sdomdomtr 9128 | . 2 ⊢ ((𝐵 ≺ ω ∧ ω ≼ 𝐴) → 𝐵 ≺ 𝐴) | |
7 | 4, 5, 6 | syl2anc 582 | 1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3463 class class class wbr 5144 ωcom 7865 ≼ cdom 8955 ≺ csdm 8956 |
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 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
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 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7866 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 |
This theorem is referenced by: (None) |
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