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Mirrors > Home > MPE Home > Th. List > infsdomnnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of infsdomnn 9319 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 8959 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5728 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ V) |
3 | nnsdomg 9316 | . . 3 ⊢ ((ω ∈ V ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) | |
4 | 2, 3 | sylan 579 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) |
5 | simpl 482 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → ω ≼ 𝐴) | |
6 | sdomdomtr 9124 | . 2 ⊢ ((𝐵 ≺ ω ∧ ω ≼ 𝐴) → 𝐵 ≺ 𝐴) | |
7 | 4, 5, 6 | syl2anc 583 | 1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 Vcvv 3469 class class class wbr 5142 ωcom 7862 ≼ cdom 8951 ≺ csdm 8952 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 |
This theorem is referenced by: (None) |
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