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Mirrors > Home > MPE Home > Th. List > unsnen | Structured version Visualization version GIF version |
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
unsnen.1 | ⊢ 𝐴 ∈ V |
unsnen.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
unsnen | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn 4649 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
2 | cardon 9375 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
3 | 2 | onordi 6297 | . . . . 5 ⊢ Ord (card‘𝐴) |
4 | orddisj 6231 | . . . . 5 ⊢ (Ord (card‘𝐴) → ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅ |
6 | unsnen.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | 6 | cardid 9971 | . . . . . 6 ⊢ (card‘𝐴) ≈ 𝐴 |
8 | 7 | ensymi 8561 | . . . . 5 ⊢ 𝐴 ≈ (card‘𝐴) |
9 | unsnen.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
10 | fvex 6685 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
11 | en2sn 8595 | . . . . . 6 ⊢ ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)}) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ {𝐵} ≈ {(card‘𝐴)} |
13 | unen 8598 | . . . . 5 ⊢ (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) | |
14 | 8, 12, 13 | mpanl12 700 | . . . 4 ⊢ (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
15 | 5, 14 | mpan2 689 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
16 | 1, 15 | sylbir 237 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
17 | df-suc 6199 | . 2 ⊢ suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)}) | |
18 | 16, 17 | breqtrrdi 5110 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 class class class wbr 5068 Ord word 6192 suc csuc 6195 ‘cfv 6357 ≈ cen 8508 cardccrd 9366 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-wrecs 7949 df-recs 8010 df-1o 8104 df-er 8291 df-en 8512 df-card 9370 df-ac 9544 |
This theorem is referenced by: (None) |
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