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Mirrors > Home > MPE Home > Th. List > ex-1st | Structured version Visualization version GIF version |
Description: Example for df-1st 7739. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
ex-1st | ⊢ (1st ‘〈3, 4〉) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ex 11877 | . 2 ⊢ 3 ∈ V | |
2 | 4re 11879 | . . 3 ⊢ 4 ∈ ℝ | |
3 | 2 | elexi 3417 | . 2 ⊢ 4 ∈ V |
4 | 1, 3 | op1st 7747 | 1 ⊢ (1st ‘〈3, 4〉) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 〈cop 4533 ‘cfv 6358 1st c1st 7737 ℝcr 10693 3c3 11851 4c4 11852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-i2m1 10762 ax-1ne0 10763 ax-rrecex 10766 ax-cnre 10767 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-1st 7739 df-2 11858 df-3 11859 df-4 11860 |
This theorem is referenced by: (None) |
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