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Mirrors > Home > MPE Home > Th. List > djudom1 | Structured version Visualization version GIF version |
Description: Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 1-Sep-2023.) |
Ref | Expression |
---|---|
djudom1 | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5325 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | xpdom2 8605 | . . 3 ⊢ (𝐴 ≼ 𝐵 → ({∅} × 𝐴) ≼ ({∅} × 𝐵)) |
3 | snex 5325 | . . . . 5 ⊢ {1o} ∈ V | |
4 | xpexg 7466 | . . . . 5 ⊢ (({1o} ∈ V ∧ 𝐶 ∈ 𝑉) → ({1o} × 𝐶) ∈ V) | |
5 | 3, 4 | mpan 688 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ({1o} × 𝐶) ∈ V) |
6 | domrefg 8537 | . . . 4 ⊢ (({1o} × 𝐶) ∈ V → ({1o} × 𝐶) ≼ ({1o} × 𝐶)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ({1o} × 𝐶) ≼ ({1o} × 𝐶)) |
8 | xp01disjl 8114 | . . . 4 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅ | |
9 | undom 8598 | . . . 4 ⊢ (((({∅} × 𝐴) ≼ ({∅} × 𝐵) ∧ ({1o} × 𝐶) ≼ ({1o} × 𝐶)) ∧ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶))) | |
10 | 8, 9 | mpan2 689 | . . 3 ⊢ ((({∅} × 𝐴) ≼ ({∅} × 𝐵) ∧ ({1o} × 𝐶) ≼ ({1o} × 𝐶)) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶))) |
11 | 2, 7, 10 | syl2an 597 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶))) |
12 | df-dju 9323 | . 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶)) | |
13 | df-dju 9323 | . 2 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶)) | |
14 | 11, 12, 13 | 3brtr4g 5093 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 ∩ cin 3928 ∅c0 4284 {csn 4560 class class class wbr 5059 × cxp 5546 1oc1o 8088 ≼ cdom 8500 ⊔ cdju 9320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1o 8095 df-en 8503 df-dom 8504 df-dju 9323 |
This theorem is referenced by: djudom2 9602 djulepw 9611 unctb 9620 infdif 9624 gchdjuidm 10083 gchpwdom 10085 gchhar 10094 pr2dom 39967 tr3dom 39968 |
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