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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33684 | . . 3 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) | |
2 | 1 | uneq2i 4050 | . 2 ⊢ (( O ‘𝐴) ∪ ( N ‘𝐴)) = (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) |
3 | oldssmade 33705 | . . 3 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) | |
4 | undif 4371 | . . 3 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴)) | |
5 | 3, 4 | mpbi 233 | . 2 ⊢ (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴) |
6 | 2, 5 | eqtr2i 2762 | 1 ⊢ ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3840 ∪ cun 3841 ⊆ wss 3843 ‘cfv 6339 M cmade 33671 O cold 33672 N cnew 33673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-wrecs 7978 df-recs 8039 df-1o 8133 df-2o 8134 df-no 33491 df-slt 33492 df-bday 33493 df-sslt 33621 df-scut 33623 df-made 33676 df-old 33677 df-new 33678 |
This theorem is referenced by: oldsuc 33713 |
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