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Mirrors > Home > MPE Home > Th. List > madeun | Structured version Visualization version GIF version |
Description: The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.) |
Ref | Expression |
---|---|
madeun | ⊢ ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | newval 27695 | . . 3 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) | |
2 | 1 | uneq2i 4160 | . 2 ⊢ (( O ‘𝐴) ∪ ( N ‘𝐴)) = (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) |
3 | oldssmade 27717 | . . 3 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) | |
4 | undif 4481 | . . 3 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴)) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴) |
6 | 2, 5 | eqtr2i 2760 | 1 ⊢ ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ‘cfv 6543 M cmade 27682 O cold 27683 N cnew 27684 |
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-rmo 3375 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-tp 4633 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-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-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-1o 8472 df-2o 8473 df-no 27489 df-slt 27490 df-bday 27491 df-sslt 27627 df-scut 27629 df-made 27687 df-old 27688 df-new 27689 |
This theorem is referenced by: oldsuc 27725 sltonold 28066 |
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