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Mirrors > Home > MPE Home > Th. List > ex-pss | Structured version Visualization version GIF version |
Description: Example for df-pss 3951. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-pss | ⊢ {1, 2} ⊊ {1, 2, 3} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-ss 28133 | . 2 ⊢ {1, 2} ⊆ {1, 2, 3} | |
2 | 3ex 11707 | . . . . 5 ⊢ 3 ∈ V | |
3 | 2 | tpid3 4701 | . . . 4 ⊢ 3 ∈ {1, 2, 3} |
4 | 1re 10629 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | 1lt3 11798 | . . . . . 6 ⊢ 1 < 3 | |
6 | 4, 5 | gtneii 10740 | . . . . 5 ⊢ 3 ≠ 1 |
7 | 2re 11699 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt3 11797 | . . . . . 6 ⊢ 2 < 3 | |
9 | 7, 8 | gtneii 10740 | . . . . 5 ⊢ 3 ≠ 2 |
10 | 6, 9 | nelpri 4584 | . . . 4 ⊢ ¬ 3 ∈ {1, 2} |
11 | nelne1 3110 | . . . 4 ⊢ ((3 ∈ {1, 2, 3} ∧ ¬ 3 ∈ {1, 2}) → {1, 2, 3} ≠ {1, 2}) | |
12 | 3, 10, 11 | mp2an 688 | . . 3 ⊢ {1, 2, 3} ≠ {1, 2} |
13 | 12 | necomi 3067 | . 2 ⊢ {1, 2} ≠ {1, 2, 3} |
14 | df-pss 3951 | . 2 ⊢ ({1, 2} ⊊ {1, 2, 3} ↔ ({1, 2} ⊆ {1, 2, 3} ∧ {1, 2} ≠ {1, 2, 3})) | |
15 | 1, 13, 14 | mpbir2an 707 | 1 ⊢ {1, 2} ⊊ {1, 2, 3} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ⊊ wpss 3934 {cpr 4559 {ctp 4561 1c1 10526 2c2 11680 3c3 11681 |
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-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-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 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-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-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-2 11688 df-3 11689 |
This theorem is referenced by: (None) |
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