Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hashrabsn1 | Structured version Visualization version GIF version |
Description: If the size of a restricted class abstraction restricted to a singleton is 1, the condition of the class abstraction must hold for the singleton. (Contributed by Alexander van der Vekens, 3-Sep-2018.) |
Ref | Expression |
---|---|
hashrabsn1 | ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} | |
2 | rabrsn 4652 | . 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} → ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})) | |
3 | fveqeq2 6672 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 ↔ (♯‘∅) = 1)) | |
4 | hash0 13716 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
5 | 4 | eqeq1i 2823 | . . . . 5 ⊢ ((♯‘∅) = 1 ↔ 0 = 1) |
6 | 0ne1 11696 | . . . . . 6 ⊢ 0 ≠ 1 | |
7 | eqneqall 3024 | . . . . . 6 ⊢ (0 = 1 → (0 ≠ 1 → [𝐴 / 𝑥]𝜑)) | |
8 | 6, 7 | mpi 20 | . . . . 5 ⊢ (0 = 1 → [𝐴 / 𝑥]𝜑) |
9 | 5, 8 | sylbi 218 | . . . 4 ⊢ ((♯‘∅) = 1 → [𝐴 / 𝑥]𝜑) |
10 | 3, 9 | syl6bi 254 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
11 | snidg 4589 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → 𝐴 ∈ {𝐴}) |
13 | eleq2 2898 | . . . . . . . . 9 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝐴 ∈ {𝐴})) | |
14 | 13 | adantl 482 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝐴 ∈ {𝐴})) |
15 | 12, 14 | mpbird 258 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → 𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑}) |
16 | nfcv 2974 | . . . . . . . . 9 ⊢ Ⅎ𝑥{𝐴} | |
17 | 16 | elrabsf 3813 | . . . . . . . 8 ⊢ (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ (𝐴 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)) |
18 | 17 | simprbi 497 | . . . . . . 7 ⊢ (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → [𝐴 / 𝑥]𝜑) |
19 | 15, 18 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → [𝐴 / 𝑥]𝜑) |
20 | 19 | a1d 25 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
21 | 20 | ex 413 | . . . 4 ⊢ (𝐴 ∈ V → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
22 | snprc 4645 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
23 | eqeq2 2830 | . . . . . 6 ⊢ ({𝐴} = ∅ → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅)) | |
24 | ax-1ne0 10594 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
25 | eqneqall 3024 | . . . . . . . . . 10 ⊢ (1 = 0 → (1 ≠ 0 → [𝐴 / 𝑥]𝜑)) | |
26 | 24, 25 | mpi 20 | . . . . . . . . 9 ⊢ (1 = 0 → [𝐴 / 𝑥]𝜑) |
27 | 26 | eqcoms 2826 | . . . . . . . 8 ⊢ (0 = 1 → [𝐴 / 𝑥]𝜑) |
28 | 5, 27 | sylbi 218 | . . . . . . 7 ⊢ ((♯‘∅) = 1 → [𝐴 / 𝑥]𝜑) |
29 | 3, 28 | syl6bi 254 | . . . . . 6 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
30 | 23, 29 | syl6bi 254 | . . . . 5 ⊢ ({𝐴} = ∅ → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
31 | 22, 30 | sylbi 218 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
32 | 21, 31 | pm2.61i 183 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
33 | 10, 32 | jaoi 851 | . 2 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
34 | 1, 2, 33 | mp2b 10 | 1 ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 [wsbc 3769 ∅c0 4288 {csn 4557 ‘cfv 6348 0cc0 10525 1c1 10526 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |