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Mirrors > Home > MPE Home > Th. List > naddel12 | Structured version Visualization version GIF version |
Description: Natural addition to both sides of ordinal less-than. (Contributed by Scott Fenton, 7-Feb-2025.) |
Ref | Expression |
---|---|
naddel12 | ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 770 | . . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐵 ∈ 𝐷) | |
2 | onelon 6389 | . . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐵 ∈ 𝐷) → 𝐵 ∈ On) | |
3 | 2 | ad2ant2l 743 | . . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐵 ∈ On) |
4 | simplr 766 | . . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐷 ∈ On) | |
5 | onelon 6389 | . . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ On) | |
6 | 5 | ad2ant2r 744 | . . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐴 ∈ On) |
7 | naddel2 8691 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐷 ↔ (𝐴 +no 𝐵) ∈ (𝐴 +no 𝐷))) | |
8 | 3, 4, 6, 7 | syl3anc 1370 | . . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐵 ∈ 𝐷 ↔ (𝐴 +no 𝐵) ∈ (𝐴 +no 𝐷))) |
9 | 1, 8 | mpbid 231 | . . 3 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴 +no 𝐵) ∈ (𝐴 +no 𝐷)) |
10 | simprl 768 | . . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐴 ∈ 𝐶) | |
11 | simpll 764 | . . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → 𝐶 ∈ On) | |
12 | naddel1 8690 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ∈ 𝐶 ↔ (𝐴 +no 𝐷) ∈ (𝐶 +no 𝐷))) | |
13 | 6, 11, 4, 12 | syl3anc 1370 | . . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴 ∈ 𝐶 ↔ (𝐴 +no 𝐷) ∈ (𝐶 +no 𝐷))) |
14 | 10, 13 | mpbid 231 | . . 3 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴 +no 𝐷) ∈ (𝐶 +no 𝐷)) |
15 | naddcl 8680 | . . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +no 𝐷) ∈ On) | |
16 | 15 | adantr 480 | . . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐶 +no 𝐷) ∈ On) |
17 | ontr1 6410 | . . . 4 ⊢ ((𝐶 +no 𝐷) ∈ On → (((𝐴 +no 𝐵) ∈ (𝐴 +no 𝐷) ∧ (𝐴 +no 𝐷) ∈ (𝐶 +no 𝐷)) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷))) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (((𝐴 +no 𝐵) ∈ (𝐴 +no 𝐷) ∧ (𝐴 +no 𝐷) ∈ (𝐶 +no 𝐷)) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷))) |
19 | 9, 14, 18 | mp2and 696 | . 2 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷)) |
20 | 19 | ex 412 | 1 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 Oncon0 6364 (class class class)co 7412 +no cnadd 8668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 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-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-frecs 8270 df-nadd 8669 |
This theorem is referenced by: mulsproplem4 27815 mulsproplem5 27816 mulsproplem6 27817 mulsproplem7 27818 mulsproplem8 27819 |
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