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Mirrors > Home > MPE Home > Th. List > upgr1elem | Structured version Visualization version GIF version |
Description: Lemma for upgr1e 28939 and uspgr1e 29070. (Contributed by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
upgr1elem.s | ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑆) |
upgr1elem.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
upgr1elem | ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 ⊢ (𝑥 = {𝐵, 𝐶} → (♯‘𝑥) = (♯‘{𝐵, 𝐶})) | |
2 | 1 | breq1d 5158 | . . 3 ⊢ (𝑥 = {𝐵, 𝐶} → ((♯‘𝑥) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) ≤ 2)) |
3 | upgr1elem.s | . . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑆) | |
4 | upgr1elem.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | prnzg 4783 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → {𝐵, 𝐶} ≠ ∅) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ≠ ∅) |
7 | eldifsn 4791 | . . . 4 ⊢ ({𝐵, 𝐶} ∈ (𝑆 ∖ {∅}) ↔ ({𝐵, 𝐶} ∈ 𝑆 ∧ {𝐵, 𝐶} ≠ ∅)) | |
8 | 3, 6, 7 | sylanbrc 582 | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ (𝑆 ∖ {∅})) |
9 | hashprlei 14462 | . . . . 5 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2) | |
10 | 9 | simpri 485 | . . . 4 ⊢ (♯‘{𝐵, 𝐶}) ≤ 2 |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → (♯‘{𝐵, 𝐶}) ≤ 2) |
12 | 2, 8, 11 | elrabd 3684 | . 2 ⊢ (𝜑 → {𝐵, 𝐶} ∈ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
13 | 12 | snssd 4813 | 1 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4323 {csn 4629 {cpr 4631 class class class wbr 5148 ‘cfv 6548 Fincfn 8964 ≤ cle 11280 2c2 12298 ♯chash 14322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-fz 13518 df-hash 14323 |
This theorem is referenced by: upgr1e 28939 uspgr1e 29070 |
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